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1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> Zariski topology </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/7704/#Item_37" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="topology">Topology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/topology">topology</a></strong> (<a class="existingWikiWord" href="/nlab/show/point-set+topology">point-set topology</a>, <a class="existingWikiWord" href="/nlab/show/point-free+topology">point-free topology</a>)</p> <p>see also <em><a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a></em>, <em><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></em>, <em><a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a></em> and <em><a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a> <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></em></p> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology">Introduction</a></p> <p><strong>Basic concepts</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a>, <a class="existingWikiWord" href="/nlab/show/closed+subset">closed subset</a>, <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+for+the+topology">base for the topology</a>, <a class="existingWikiWord" href="/nlab/show/neighbourhood+base">neighbourhood base</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finer+topology">finer/coarser topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+closure">closure</a>, <a class="existingWikiWord" href="/nlab/show/topological+interior">interior</a>, <a class="existingWikiWord" href="/nlab/show/topological+boundary">boundary</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/separation+axiom">separation</a>, <a class="existingWikiWord" href="/nlab/show/sober+topological+space">sobriety</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a>, <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/uniformly+continuous+function">uniformly continuous function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+embedding">embedding</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+map">open map</a>, <a class="existingWikiWord" href="/nlab/show/closed+map">closed map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequence">sequence</a>, <a class="existingWikiWord" href="/nlab/show/net">net</a>, <a class="existingWikiWord" href="/nlab/show/sub-net">sub-net</a>, <a class="existingWikiWord" href="/nlab/show/filter">filter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/convergence">convergence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a><a class="existingWikiWord" href="/nlab/show/Top">Top</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/convenient+category+of+topological+spaces">convenient category of topological spaces</a></li> </ul> </li> </ul> <p><strong><a href="Top#UniversalConstructions">Universal constructions</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/initial+topology">initial topology</a>, <a class="existingWikiWord" href="/nlab/show/final+topology">final topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/subspace">subspace</a>, <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a>,</p> </li> <li> <p>fiber space, <a class="existingWikiWord" href="/nlab/show/space+attachment">space attachment</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/product+space">product space</a>, <a class="existingWikiWord" href="/nlab/show/disjoint+union+space">disjoint union space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cylinder">mapping cylinder</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocylinder">mapping cocylinder</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+telescope">mapping telescope</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/colimits+of+normal+spaces">colimits of normal spaces</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">Extra stuff, structure, properties</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/nice+topological+space">nice topological space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>, <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a>, <a class="existingWikiWord" href="/nlab/show/metrisable+space">metrisable space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kolmogorov+space">Kolmogorov space</a>, <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff space</a>, <a class="existingWikiWord" href="/nlab/show/regular+space">regular space</a>, <a class="existingWikiWord" href="/nlab/show/normal+space">normal space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sober+space">sober space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+space">compact space</a>, <a class="existingWikiWord" href="/nlab/show/proper+map">proper map</a></p> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+topological+space">sequentially compact</a>, <a class="existingWikiWord" href="/nlab/show/countably+compact+topological+space">countably compact</a>, <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+space">locally compact</a>, <a class="existingWikiWord" href="/nlab/show/sigma-compact+topological+space">sigma-compact</a>, <a class="existingWikiWord" href="/nlab/show/paracompact+space">paracompact</a>, <a class="existingWikiWord" href="/nlab/show/countably+paracompact+topological+space">countably paracompact</a>, <a class="existingWikiWord" href="/nlab/show/strongly+compact+topological+space">strongly compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compactly+generated+space">compactly generated space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second-countable+space">second-countable space</a>, <a class="existingWikiWord" href="/nlab/show/first-countable+space">first-countable space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/contractible+space">contractible space</a>, <a class="existingWikiWord" href="/nlab/show/locally+contractible+space">locally contractible space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+space">connected space</a>, <a class="existingWikiWord" href="/nlab/show/locally+connected+space">locally connected space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simply-connected+space">simply-connected space</a>, <a class="existingWikiWord" href="/nlab/show/locally+simply-connected+space">locally simply-connected space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a>, <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a>, <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a>, <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/empty+space">empty space</a>, <a class="existingWikiWord" href="/nlab/show/point+space">point space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+space">discrete space</a>, <a class="existingWikiWord" href="/nlab/show/codiscrete+space">codiscrete space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sierpinski+space">Sierpinski space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/order+topology">order topology</a>, <a class="existingWikiWord" href="/nlab/show/specialization+topology">specialization topology</a>, <a class="existingWikiWord" href="/nlab/show/Scott+topology">Scott topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/real+line">real line</a>, <a class="existingWikiWord" href="/nlab/show/plane">plane</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder">cylinder</a>, <a class="existingWikiWord" href="/nlab/show/cone">cone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere">sphere</a>, <a class="existingWikiWord" href="/nlab/show/ball">ball</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle">circle</a>, <a class="existingWikiWord" href="/nlab/show/torus">torus</a>, <a class="existingWikiWord" href="/nlab/show/annulus">annulus</a>, <a class="existingWikiWord" href="/nlab/show/Moebius+strip">Moebius strip</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polytope">polytope</a>, <a class="existingWikiWord" href="/nlab/show/polyhedron">polyhedron</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/projective+space">projective space</a> (<a class="existingWikiWord" href="/nlab/show/real+projective+space">real</a>, <a class="existingWikiWord" href="/nlab/show/complex+projective+space">complex</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/configuration+space+%28mathematics%29">configuration space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path">path</a>, <a class="existingWikiWord" href="/nlab/show/loop">loop</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+spaces">mapping spaces</a>: <a class="existingWikiWord" href="/nlab/show/compact-open+topology">compact-open topology</a>, <a class="existingWikiWord" href="/nlab/show/topology+of+uniform+convergence">topology of uniform convergence</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a>, <a class="existingWikiWord" href="/nlab/show/path+space">path space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Zariski+topology">Zariski topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cantor+space">Cantor space</a>, <a class="existingWikiWord" href="/nlab/show/Mandelbrot+space">Mandelbrot space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Peano+curve">Peano curve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/line+with+two+origins">line with two origins</a>, <a class="existingWikiWord" href="/nlab/show/long+line">long line</a>, <a class="existingWikiWord" href="/nlab/show/Sorgenfrey+line">Sorgenfrey line</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-topology">K-topology</a>, <a class="existingWikiWord" href="/nlab/show/Dowker+space">Dowker space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Warsaw+circle">Warsaw circle</a>, <a class="existingWikiWord" href="/nlab/show/Hawaiian+earring+space">Hawaiian earring space</a></p> </li> </ul> <p><strong>Basic statements</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hausdorff+spaces+are+sober">Hausdorff spaces are sober</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/schemes+are+sober">schemes are sober</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+images+of+compact+spaces+are+compact">continuous images of compact spaces are compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+subspaces+of+compact+Hausdorff+spaces+are+equivalently+compact+subspaces">closed subspaces of compact Hausdorff spaces are equivalently compact subspaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+subspaces+of+compact+Hausdorff+spaces+are+locally+compact">open subspaces of compact Hausdorff spaces are locally compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quotient+projections+out+of+compact+Hausdorff+spaces+are+closed+precisely+if+the+codomain+is+Hausdorff">quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+spaces+equivalently+have+converging+subnet+of+every+net">compact spaces equivalently have converging subnet of every net</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lebesgue+number+lemma">Lebesgue number lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+metric+spaces+are+equivalently+compact+metric+spaces">sequentially compact metric spaces are equivalently compact metric spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+spaces+equivalently+have+converging+subnet+of+every+net">compact spaces equivalently have converging subnet of every net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+metric+spaces+are+totally+bounded">sequentially compact metric spaces are totally bounded</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+metric+space+valued+function+on+compact+metric+space+is+uniformly+continuous">continuous metric space valued function on compact metric space is uniformly continuous</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+spaces+are+normal">paracompact Hausdorff spaces are normal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+spaces+equivalently+admit+subordinate+partitions+of+unity">paracompact Hausdorff spaces equivalently admit subordinate partitions of unity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+injections+are+embeddings">closed injections are embeddings</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+maps+to+locally+compact+spaces+are+closed">proper maps to locally compact spaces are closed</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+proper+maps+to+locally+compact+spaces+are+equivalently+the+closed+embeddings">injective proper maps to locally compact spaces are equivalently the closed embeddings</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+compact+and+sigma-compact+spaces+are+paracompact">locally compact and sigma-compact spaces are paracompact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+compact+and+second-countable+spaces+are+sigma-compact">locally compact and second-countable spaces are sigma-compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second-countable+regular+spaces+are+paracompact">second-countable regular spaces are paracompact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CW-complexes+are+paracompact+Hausdorff+spaces">CW-complexes are paracompact Hausdorff spaces</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Urysohn%27s+lemma">Urysohn's lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tietze+extension+theorem">Tietze extension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tychonoff+theorem">Tychonoff theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tube+lemma">tube lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael%27s+theorem">Michael's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brouwer%27s+fixed+point+theorem">Brouwer's fixed point theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+invariance+of+dimension">topological invariance of dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jordan+curve+theorem">Jordan curve theorem</a></p> </li> </ul> <p><strong>Analysis Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Heine-Borel+theorem">Heine-Borel theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/intermediate+value+theorem">intermediate value theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extreme+value+theorem">extreme value theorem</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological homotopy theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a>, <a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>, <a class="existingWikiWord" href="/nlab/show/deformation+retract">deformation retract</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a>, <a class="existingWikiWord" href="/nlab/show/covering+space">covering space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+extension+property">homotopy extension property</a>, <a class="existingWikiWord" href="/nlab/show/Hurewicz+cofibration">Hurewicz cofibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+cofiber+sequence">cofiber sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Str%C3%B8m+model+category">Strøm model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a></p> </li> </ul> </div></div> <h4 id="algebra">Algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a></p> <h2 id="algebraic_theories">Algebraic theories</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+theory">algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/2-algebraic+theory">2-algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-algebraic+theory">(∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad">monad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-monad">(∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operad">operad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-operad">(∞,1)-operad</a></p> </li> </ul> <h2 id="algebras_and_modules">Algebras and modules</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+a+monad">algebra over a monad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-monad">∞-algebra over an (∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+algebraic+theory">algebra over an algebraic theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-algebraic+theory">∞-algebra over an (∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over an operad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-operad">∞-algebra over an (∞,1)-operad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a></p> </li> </ul> <h2 id="higher_algebras">Higher algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+in+an+%28%E2%88%9E%2C1%29-category">monoid in an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+an+%28%E2%88%9E%2C1%29-category">commutative monoid in an (∞,1)-category</a></p> </li> </ul> </li> <li> <p>symmetric monoidal (∞,1)-category of spectra</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+smash+product+of+spectra">symmetric monoidal smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</a>, <a class="existingWikiWord" href="/nlab/show/module+spectrum">module spectrum</a>, <a class="existingWikiWord" href="/nlab/show/algebra+spectrum">algebra spectrum</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebra">A-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+space">A-∞ space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/C-%E2%88%9E+algebra">C-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+algebra">E-∞ algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module+bundle">(∞,1)-module bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative cohomology theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/deformation+theory">deformation theory</a></li> </ul> </li> </ul> <h2 id="model_category_presentations">Model category presentations</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+T-algebras">model structure on simplicial T-algebras</a> / <a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+operads">model structure on operads</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">model structure on algebras over an operad</a></p> </li> </ul> <h2 id="geometry_on_formal_duals_of_algebras">Geometry on formal duals of algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+conjecture">Deligne conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/higher+algebra+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#OnAffineSpace'>On affine space</a></li> <ul> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#topological_closures'>Topological closures</a></li> <li><a href='#irreducible_closed_subsets_as_prime_ideals'>Irreducible closed subsets as prime ideals</a></li> </ul> <li><a href='#examples'>Examples</a></li> </ul> <li><a href='#OnAffineVarieties'>On affine varieties</a></li> <ul> <li><a href='#definition_4'>Definition</a></li> <li><a href='#PropertiesOnAffineVarieties'>Properties</a></li> <ul> <li><a href='#topological_closures_2'>Topological closures</a></li> <li><a href='#irreducible_closed_subsets_as_prime_ideals_2'>Irreducible closed subsets as prime ideals</a></li> </ul> <li><a href='#examples_2'>Examples</a></li> </ul> <li><a href='#InTermsOfGaloisConnections'>In terms of Galois connections</a></li> <ul> <li><a href='#BackgroundOnGaloisConnections'>Background on Galois connections</a></li> <li><a href='#GaloisConnectionAppliedToAffineSpace'>Applied to affine space</a></li> <li><a href='#GaloisAppliedToAffineSchemes'>Applied to affine schemes</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The <em>Zariski topology</em> is a <a class="existingWikiWord" href="/nlab/show/topological+space">topology</a> on the <a class="existingWikiWord" href="/nlab/show/prime+spectrum+of+a+commutative+ring">prime spectrum of a commutative ring</a>. It serves as the basis for much of <a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a>.</p> <p id="Outline"> We consider the definition in increasing generality and sophistication:</p> <ol> <li> <p>First we discuss the naive Zariski topology <a href="#OnAffineSpace">on affine spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">k^n</annotation></semantics></math>, consider the classical proofs and discover thereby the special role of prime ideals and maximal ideals;</p> </li> <li> <p>then we turn to the modern definition of the Zariski topology <a href="#OnAffineVarieties">on affine varieties</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(R)</annotation></semantics></math> which takes the concept of prime (and maximal) ideals as primary, and again we provide the classical arguments;</p> </li> <li> <p>finally we discuss the abstract <a class="existingWikiWord" href="/nlab/show/category+theory">category theoretic</a> perspective on these matters <a href="#InTermsOfGaloisConnections">in terms of Galois connections</a> and obtain slick category theoretic proofs of all the previous statements.</p> </li> </ol> <p>Starting with <a class="existingWikiWord" href="/nlab/show/affine+space">affine space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">k^n</annotation></semantics></math>, then the idea of the Zariski topology is to take as the <em><a class="existingWikiWord" href="/nlab/show/closed+subsets">closed subsets</a></em> those defined by the vanishing of any set of <a class="existingWikiWord" href="/nlab/show/polynomials">polynomials</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/variables">variables</a>, hence the solution sets to <a class="existingWikiWord" href="/nlab/show/equations">equations</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo>∀</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mrow><mo>(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \underset{i \in I}{\forall} \left( f_i(x_1, \cdots, x_n) = 0 \right) </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub><mo>∈</mo><mi>k</mi><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">f_i \in k[X_1, \cdots, X_n]</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/polynomials">polynomials</a>. The <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a> of the topology are the <a class="existingWikiWord" href="/nlab/show/complements">complements</a> of these <em>vanishing sets</em>.</p> <p>It is clear that the vanishing set such a set of polynomials depends only on the <a class="existingWikiWord" href="/nlab/show/ideal">ideal</a> in the <a class="existingWikiWord" href="/nlab/show/polynomial+ring">polynomial ring</a> which is generated by them. Under this translation then forming the intersection of closed subsets corresponds to forming the sum of these ideals, and forming the union of closed subsets corresponds to forming the product of the corresponding ideals. This way the Zariski topology establishes a dictionary between topological concepts of the affine space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">k^n</annotation></semantics></math>, and algebra inside the polynomial ring.</p> <p>In particular one finds that the <a class="existingWikiWord" href="/nlab/show/irreducible+closed+subsets">irreducible closed subsets</a> of the Zariski topology correspond to the <a class="existingWikiWord" href="/nlab/show/prime+ideals">prime ideals</a> in the polynomial ring (prop. <a class="maruku-ref" href="#PrimeVanishingIdealOfIrreducibleZariskiClosed"></a> and prop. <a class="maruku-ref" href="#PrimeIdealClosedsubspaceBijection"></a> below), and that the <a class="existingWikiWord" href="/nlab/show/closed+points">closed points</a> correspond to the <a class="existingWikiWord" href="/nlab/show/maximal+ideals">maximal ideals</a> among these (prop. <a class="maruku-ref" href="#MaximalIdealsAreClosedPoints"></a>).</p> <p>This motivates the modern refinement of the concept of the Zariski topology, where one considers any <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> and equips its set of <a class="existingWikiWord" href="/nlab/show/prime+ideals">prime ideals</a> with a topology, by direct analogy with the previously naive affine space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">k^n</annotation></semantics></math>, which is recovered with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> a polynomial ring and restricting attention to the maximal ideals (example <a class="maruku-ref" href="#AffinSpaceAsPrimeSpectrum"></a> below).</p> <p>These sets of prime ideals of a ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> equipped with the Zariski topology are called the (topological spaces underlying) the <em><a class="existingWikiWord" href="/nlab/show/prime+spectrum+of+a+commutative+ring">prime spectrum of a commutative ring</a></em>, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(R)</annotation></semantics></math>.</p> <p>The Zariski topology is in general not <a class="existingWikiWord" href="/nlab/show/Hausdorff+topological+space">Hausdorff</a> (example <a class="maruku-ref" href="#AffineSpaceOverInfiniteFieldNotHausdorff"></a> below) which makes it sometimes be regarded as an “exotic” type of topology. But it is in fact <a class="existingWikiWord" href="/nlab/show/sober+topological+space">sober</a> (prop. <a class="maruku-ref" href="#ZariskiTopologyIsSober"></a> below) and hence as well-behaved in this respect as general <a class="existingWikiWord" href="/nlab/show/locales">locales</a> are.</p> <h2 id="OnAffineSpace">On affine space</h2> <p>We consider here, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/field">field</a>, the <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">k^n</annotation></semantics></math> equipped with a Zariski topology. This is the original definition of Zariski topology, and serves well to motivate the concept, but eventually it was superceded by a more refined concept of Zariski topologies of prime spectra, discussed in the next subsection <a href="#OnAffineVarieties">below</a>. In example <a class="maruku-ref" href="#AffinSpaceAsPrimeSpectrum"></a> below we reconsider the naive case of interest in this subsection here from that more refined perspective.</p> <h3 id="definition">Definition</h3> <div class="num_defn" id="ZariskiOpenSubsetsOnAffineSpace"> <h6 id="definition_2">Definition</h6> <p><strong>(Zariski topology on affine space)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/field">field</a>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">k[X_1, \cdots, X_n]</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/set">set</a> of <a class="existingWikiWord" href="/nlab/show/polynomials">polynomials</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/variables">variables</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℱ</mi><mo>⊂</mo><mi>k</mi><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathcal{F} \subset k[X_1, \cdots, X_n]</annotation></semantics></math> a subset of polynomials, let the subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo>⊂</mo><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">V(\mathcal{F}) \subset k^n</annotation></semantics></math> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-fold <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> of the underlying set of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> (the <em>vanishing set</em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℱ</mi></mrow><annotation encoding="application/x-tex">\mathcal{F}</annotation></semantics></math>) be the subset of points on which all these polynomials jointly vanish:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo>≔</mo><mrow><mo>{</mo><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>k</mi> <mi>n</mi></msup><mspace width="thinmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thinmathspace"></mspace><munder><mo>∀</mo><mrow><mi>f</mi><mo>∈</mo><mi>ℱ</mi></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> V(\mathcal{F}) \coloneqq \left\{ (a_1, \cdots, a_n) \in k^n \,\vert\, \underset{f \in \mathcal{F}}{\forall} f(a_1, \cdots, a_n) = 0 \right\} \,. </annotation></semantics></math></div> <p>These subsets are called the <em>Zariski <a class="existingWikiWord" href="/nlab/show/closed+subsets">closed subsets</a></em>.</p> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mrow><msubsup><mi>𝔸</mi> <mi>k</mi> <mi>n</mi></msubsup></mrow></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mrow><mo>{</mo><msup><mi>k</mi> <mi>n</mi></msup><mo>\</mo><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo>⊂</mo><msup><mi>k</mi> <mi>n</mi></msup><mspace width="thinmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thinmathspace"></mspace><mi>ℱ</mi><mo>⊂</mo><mi>k</mi><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">]</mo><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> \tau_{\mathbb{A}^n_k} \;\coloneqq\; \left\{ k^n \backslash V(\mathcal{F}) \subset k^n \,\vert\, \mathcal{F} \subset k[X_1, \cdots, X_n] \right\} </annotation></semantics></math></div> <p>for the set of <a class="existingWikiWord" href="/nlab/show/complements">complements</a> of the Zariski closed subsets. These are called the <em>Zariski <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a></em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">k^n</annotation></semantics></math>.</p> </div> <div class="num_prop" id="VerifyingZariskiTopologyOnAffineSpace"> <h6 id="proposition">Proposition</h6> <p><strong>(Zariski topology is well defined)</strong></p> <p>Assuming <a class="existingWikiWord" href="/nlab/show/excluded+middle">excluded middle</a>, then:</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/field">field</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, then the Zariski open subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">k^n</annotation></semantics></math> (def. <a class="maruku-ref" href="#ZariskiOpenSubsetsOnAffineSpace"></a>) form a <a class="existingWikiWord" href="/nlab/show/topological+space">topology</a>. The resulting <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝔸</mi> <mi>k</mi> <mi>n</mi></msubsup><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><msup><mi>k</mi> <mi>n</mi></msup><mo>,</mo><msub><mi>τ</mi> <mrow><msubsup><mi>𝔸</mi> <mi>k</mi> <mi>n</mi></msubsup></mrow></msub><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \mathbb{A}^n_k \;\coloneqq\; \left( k^n, \tau_{\mathbb{A}^n_k} \right) </annotation></semantics></math></div> <p>is also called the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-dimensional <em><a class="existingWikiWord" href="/nlab/show/affine+space">affine space</a></em> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>We need to show for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>ℱ</mi> <mi>i</mi></msub><mo>⊂</mo><mi>k</mi><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">]</mo><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{\mathcal{F}_i \subset k[X_1, \cdots, X_n]\}_{i \in I}</annotation></semantics></math> a set of subsets of polynomials that</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mo>∪</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mrow><mo>(</mo><msup><mi>k</mi> <mi>n</mi></msup><mo>\</mo><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>=</mo><msup><mi>k</mi> <mi>n</mi></msup><mo>\</mo><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mo>∪</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\underset{i \in I}{\cup} \left(k^n \backslash V(\mathcal{F}_i)\right) = k^n \backslash V(\mathcal{F}_\cup)</annotation></semantics></math> for some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℱ</mi> <mo>∪</mo></msub><mo>⊂</mo><mi>k</mi><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathcal{F}_\cup \subset k[X_1, \cdots, X_n]</annotation></semantics></math>;</p> </li> <li> <p>if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/finite+set">finite</a> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mo>∩</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mrow><mo>(</mo><msup><mi>k</mi> <mi>n</mi></msup><mo>\</mo><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mo>∩</mo></msub><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>=</mo><msup><mi>k</mi> <mi>n</mi></msup><mo>\</mo><msub><mi>ℱ</mi> <mo>∩</mo></msub></mrow><annotation encoding="application/x-tex">\underset{i \in I}{\cap} \left( k^n \backslash V(\mathcal{F}_{\cap})\right) = k^n \backslash \mathcal{F}_\cap</annotation></semantics></math> for some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℱ</mi> <mo>∩</mo></msub><mo>⊂</mo><mi>k</mi><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathcal{F}_{\cap} \subset k[X_1, \cdots, X_n]</annotation></semantics></math>.</p> </li> </ol> <p>By <a class="existingWikiWord" href="/nlab/show/de+Morgan%27s+law">de Morgan's law</a> for <a class="existingWikiWord" href="/nlab/show/complements">complements</a> (and using <a class="existingWikiWord" href="/nlab/show/excluded+middle">excluded middle</a>) this is equivalent to</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mo>∩</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mo>∪</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\underset{i \in I}{\cap} V(\mathcal{F}_i) = V(\mathcal{F}_\cup)</annotation></semantics></math> for some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℱ</mi> <mo>∪</mo></msub><mo>⊂</mo><mi>k</mi><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathcal{F}_\cup \subset k[X_1, \cdots, X_n]</annotation></semantics></math>;</p> </li> <li> <p>if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/finite+set">finite</a> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mo>∪</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mo>∩</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\underset{i \in I}{\cup} V(\mathcal{F}_i) = V(\mathcal{F}_{\cap})</annotation></semantics></math> for some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℱ</mi> <mo>∩</mo></msub><mo>⊂</mo><mi>k</mi><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathcal{F}_{\cap} \subset k[X_1, \cdots, X_n]</annotation></semantics></math>.</p> </li> </ol> <p>We claim that we may take</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℱ</mi> <mo>∪</mo></msub><mo>=</mo><munder><mo>∪</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>ℱ</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{F}_\cup = \underset{i \in I}{\cup} \mathcal{F}_i</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℱ</mi> <mo>∩</mo></msub><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>ℱ</mi> <mi>i</mi></msub><mo>≔</mo><mrow><mo>{</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>f</mi> <mi>i</mi></msub><mspace width="thinmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thinmathspace"></mspace><msub><mi>f</mi> <mi>i</mi></msub><mo>∈</mo><msub><mi>ℱ</mi> <mi>i</mi></msub><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex">\mathcal{F}_{\cap} = \underset{i \in I}{\prod} \mathcal{F}_i \coloneqq \left\{ \underset{i \in I}{\prod} f_i \,\vert\, f_i \in \mathcal{F}_i \right\}</annotation></semantics></math>.</p> </li> </ol> <p>(In the second line we have the set of all those polynomials which arise as products of polynomials with one factor from each of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℱ</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{F}_i</annotation></semantics></math>.)</p> <p>Regarding the first point:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd></mtd> <mtd><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><munder><mo>∩</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thickmathspace"></mspace></mtd> <mtd><munder><mo>∀</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mrow><mo>(</mo><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>)</mo></mrow></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thickmathspace"></mspace></mtd> <mtd><munder><mo>∀</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mrow><mo>(</mo><munder><mo>∀</mo><mrow><mi>f</mi><mo>∈</mo><msub><mi>ℱ</mi> <mi>i</mi></msub></mrow></munder><mrow><mo>(</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>)</mo></mrow><mo>)</mo></mrow></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thickmathspace"></mspace></mtd> <mtd><munder><mo>∀</mo><mrow><mi>f</mi><mo>∈</mo><munder><mo>∪</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>ℱ</mi> <mi>i</mi></msub></mrow></munder><mrow><mo>(</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>)</mo></mrow></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thickmathspace"></mspace></mtd> <mtd><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><munder><mo>∪</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>ℱ</mi> <mi>i</mi></msub><mo>)</mo></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} & (a_1, \cdots, a_n) \in \underset{i \in I}{\cap} V(\mathcal{F}_i) \\ \Leftrightarrow\; & \underset{i \in I}{\forall} \left( (a_1, \cdots, a_n) \in V(\mathcal{F}_i) \right) \\ \Leftrightarrow\; & \underset{i \in I}{\forall} \left( \underset{f \in \mathcal{F}_i}{\forall} \left( f(a_1, \cdots, a_n) = 0 \right) \right) \\ \Leftrightarrow\; & \underset{f \in \underset{i \in I}{\cup} \mathcal{F}_i}{\forall} \left( f(a_1, \cdots, a_n) = 0 \right) \\ \Leftrightarrow\; & (a_1, \cdots, a_n) \in V\left( \underset{i \in I}{\cup} \mathcal{F}_i \right) \end{aligned} </annotation></semantics></math></div> <p>Regarding the second point, in one direction we have the immediate implication</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd></mtd> <mtd><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><munder><mo>∪</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thickmathspace"></mspace></mtd> <mtd><munder><mo>∃</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mrow><mo>(</mo><munder><mo>∀</mo><mrow><mi>f</mi><mo>∈</mo><msub><mi>ℱ</mi> <mi>i</mi></msub></mrow></munder><mrow><mo>(</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>)</mo></mrow><mo>)</mo></mrow></mtd></mtr> <mtr><mtd><mo>⇒</mo><mspace width="thickmathspace"></mspace></mtd> <mtd><munder><mo>∀</mo><mrow><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>ℱ</mi> <mi>i</mi></msub></mrow></munder><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>)</mo></mrow></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thickmathspace"></mspace></mtd> <mtd><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>ℱ</mi> <mi>i</mi></msub><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} & (a_1, \cdots, a_n) \in \underset{i \in I}{\cup} V(\mathcal{F}_i) \\ \Leftrightarrow\; & \underset{i \in I}{\exists} \left( \underset{f \in \mathcal{F}_i}{\forall} \left( f(a_1, \cdots, a_n) = 0 \right) \right) \\ \Rightarrow \; & \underset{(f_i) \in \underset{i \in I}{\prod} \mathcal{F}_i}{\forall} \left( \underset{i \in I}{\prod} f_i(a_1, \cdots, a_n) = 0 \right) \\ \Leftrightarrow\; & (a_1, \cdots, a_n) \in V\left( \underset{i \in I}{\prod} \mathcal{F}_i \right) \,. \end{aligned} </annotation></semantics></math></div> <p>For the converse direction we need to show that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>ℱ</mi> <mi>i</mi></msub><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>⇒</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><munder><mo>∪</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( (a_1 , \cdots , a_n) \in V\left( \underset{i \in I}{\prod} \mathcal{F}_i \right) \right) \;\Rightarrow\; \left( (a_1, \cdots, a_n) \in \underset{i \in I}{\cup} V(\mathcal{F}_1) \right) \,. </annotation></semantics></math></div> <p>hence that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><munder><mo>∀</mo><mrow><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>ℱ</mi> <mi>i</mi></msub></mrow></munder><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>⇒</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><munder><mo>∃</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mrow><mo>(</mo><munder><mo>∀</mo><mrow><msub><mi>f</mi> <mi>i</mi></msub><mo>∈</mo><msub><mi>ℱ</mi> <mi>i</mi></msub></mrow></munder><mrow><mo>(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>)</mo></mrow><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( \underset{(f_i) \in \underset{i \in I}{\prod} \mathcal{F}_i }{\forall} \left( \underset{i \in I}{\prod} f_i(a_1, \cdots, a_n) = 0 \right) \right) \;\Rightarrow\; \left( \underset{i \in I}{\exists} \left( \underset{f_i \in \mathcal{F}_i}{\forall} \left( f_i(a_1, \cdots, a_n) = 0 \right) \right) \right) \,. </annotation></semantics></math></div> <p>By <a class="existingWikiWord" href="/nlab/show/excluded+middle">excluded middle</a>, this is equivalent to its <a class="existingWikiWord" href="/nlab/show/contraposition">contraposition</a>, which by <a class="existingWikiWord" href="/nlab/show/de+Morgan%27s+law">de Morgan's law</a> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><munder><mo>∀</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mrow><mo>(</mo><munder><mo>∃</mo><mrow><msub><mi>f</mi> <mi>i</mi></msub><mo>∈</mo><msub><mi>ℱ</mi> <mi>i</mi></msub></mrow></munder><mrow><mo>(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>≠</mo><mn>0</mn><mo>)</mo></mrow><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>⇒</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><munder><mo>∃</mo><mrow><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>ℱ</mi> <mi>i</mi></msub></mrow></munder><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>≠</mo><mn>0</mn><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( \underset{i \in I}{\forall} \left( \underset{f_i \in \mathcal{F}_i}{\exists} \left( f_i(a_1, \cdots, a_n) \neq 0 \right) \right) \right) \;\Rightarrow\; \left( \underset{(f_i) \in \underset{i \in I}{\prod} \mathcal{F}_i }{\exists} \left( \underset{i \in I}{\prod} f_i(a_1, \cdots, a_n) \neq 0 \right) \right) \,. </annotation></semantics></math></div> <p>This now is true by the assumption that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/field">field</a>: If all factors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">f_i(a_1, \dots a_n) \in k</annotation></semantics></math> are non-zero, then their product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">\underset{i \in I}{\prod} f_i(a_1, \cdots, a_n) \in k</annotation></semantics></math> is non-zero.</p> </div> <h3 id="properties">Properties</h3> <h4 id="topological_closures">Topological closures</h4> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/field">field</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, consider a <a class="existingWikiWord" href="/nlab/show/subset">subset</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>⊂</mo><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex"> S \subset k^n </annotation></semantics></math></div> <p>of the underlying set of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-fold <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> with itself. Then the <a class="existingWikiWord" href="/nlab/show/topological+closure">topological closure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cl</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cl(S)</annotation></semantics></math> of this subset with respect to the Zariski topology <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mrow><msubsup><mi>𝔸</mi> <mi>k</mi> <mi>n</mi></msubsup></mrow></msub></mrow><annotation encoding="application/x-tex">\tau_{\mathbb{A}^n_k}</annotation></semantics></math> (def. <a class="maruku-ref" href="#ZariskiOpenSubsetsOnAffineSpace"></a>) is the vanishing set of all those polynomials that vanish on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Cl</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>V</mi><mrow><mo>(</mo><mrow><mo>{</mo><mi>f</mi><mo>∈</mo><mi>k</mi><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thinmathspace"></mspace><munder><mo>∀</mo><mrow><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><mi>S</mi></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>}</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Cl(S) \;=\; V \left( \left\{ f \in k[X_1, \cdots, X_n] \,\vert\, \underset{(a_1, \cdots, a_n) \in S}{\forall} f(a_1, \cdots, a_n) = 0 \right\} \right) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>We compute as follows:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>Cl</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≔</mo><munder><mo>∩</mo><mfrac linethickness="0"><mrow><mrow><mi>C</mi><mo>⊂</mo><msup><mi>k</mi> <mi>n</mi></msup><mspace width="thinmathspace"></mspace><mtext>closed</mtext></mrow></mrow><mrow><mrow><mi>C</mi><mo>⊃</mo><mi>S</mi></mrow></mrow></mfrac></munder><mi>C</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><munder><mo>∩</mo><mfrac linethickness="0"><mrow><mrow><mi>ℱ</mi><mo>⊂</mo><mi>k</mi><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">]</mo></mrow></mrow><mrow><mrow><mi>S</mi><mo>⊂</mo><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo></mrow></mrow></mfrac></munder><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>V</mi><mrow><mo>(</mo><munder><mo>∪</mo><mfrac linethickness="0"><mrow><mrow><mi>ℱ</mi><mo>⊂</mo><mi>k</mi><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">]</mo></mrow></mrow><mrow><mrow><mi>S</mi><mo>⊂</mo><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo></mrow></mrow></mfrac></munder><mi>ℱ</mi><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>V</mi><mrow><mo>(</mo><mrow><mo>{</mo><mi>f</mi><mo>∈</mo><mi>k</mi><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thinmathspace"></mspace><munder><mo>∀</mo><mrow><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><mi>S</mi></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>}</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} Cl(S) & \coloneqq \underset{ {C \subset k^n \, \text{closed}} \atop {C \supset S} }{\cap} C \\ & = \underset{ { \mathcal{F} \subset k[X_1, \cdots, X_n] } \atop { S \subset V(\mathcal{F}) } }{\cap} V(\mathcal{F}) \\ & = V\left( \underset{ { \mathcal{F} \subset k[X_1, \cdots, X_n] } \atop { S \subset V(\mathcal{F}) } }{\cup} \mathcal{F} \right) \\ & = V \left( \left\{ f \in k[X_1, \cdots, X_n] \,\vert\, \underset{(a_1, \cdots, a_n) \in S}{\forall} f(a_1, \cdots, a_n) = 0 \right\} \right) \,. \end{aligned} </annotation></semantics></math></div> <p>Here the first equality is the definition of <a class="existingWikiWord" href="/nlab/show/topological+closure">topological closure</a>, the second is the definition of closed subsets in the Zariski topology (def. <a class="maruku-ref" href="#ZariskiOpenSubsetsOnAffineSpace"></a>), the third is the expression of intersections of these in terms of unions of polynomials as in the proof of prop. <a class="maruku-ref" href="#VerifyingZariskiTopologyOnAffineSpace"></a>, and then the last one is immediate.</p> </div> <h4 id="irreducible_closed_subsets_as_prime_ideals">Irreducible closed subsets as prime ideals</h4> <p>In every <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> the <a class="existingWikiWord" href="/nlab/show/irreducible+closed+subsets">irreducible closed subsets</a> play a special role, as being precisely the points in the space as seen in its incarnation as a <a class="existingWikiWord" href="/nlab/show/locale">locale</a> (<a href="irreducible%20closed%20subspace#FrameHomomorphismsToPointAreIrrClSub">this prop.</a>). The following shows that in the Zariski topology the irreducible closed subsets all come from <a class="existingWikiWord" href="/nlab/show/prime+ideals">prime ideals</a> in the corresponding <a class="existingWikiWord" href="/nlab/show/polynomial+ring">polynomial ring</a>, and that when the <a class="existingWikiWord" href="/nlab/show/ground+field">ground field</a> is <a class="existingWikiWord" href="/nlab/show/algebraically+closed+field">algebraically closed</a>, then they are in fact in bijection to the prime ideals. See also at <em><a class="existingWikiWord" href="/nlab/show/schemes+are+sober">schemes are sober</a></em>.</p> <div class="num_defn" id="VanishingIdeal"> <h6 id="definition_3">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/vanishing+ideal">vanishing ideal</a> of Zariski closed subset)</strong>#</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/field">field</a>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>. Then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo>⊂</mo><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">V(\mathcal{F}) \subset k^n</annotation></semantics></math> a Zariski closed subset, according to def. <a class="maruku-ref" href="#ZariskiOpenSubsetsOnAffineSpace"></a>, hence for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℱ</mi><mo>⊂</mo><mi>k</mi><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathcal{F} \subset k[X_1, \cdots, X_n]</annotation></semantics></math> a set of polynomials, write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⊂</mo><mi>k</mi><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> I(V(\mathcal{F})) \subset k[X_1, \cdots, X_n] </annotation></semantics></math></div> <p>for the maximal subset of polynomials that still has the same joint vanishing set:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mrow><mo>{</mo><mi>f</mi><mo>∈</mo><mi>k</mi><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thinmathspace"></mspace><munder><mo>∀</mo><mrow><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> I(V(\mathcal{F})) \;\coloneqq\; \left\{ f \in k[X_1, \cdots, X_n] \,\vert\, \underset{(a_1, \cdots, a_n) \in V(\mathcal{F})}{\forall} f(a_1, \cdots, a_n) = 0 \right\} \,. </annotation></semantics></math></div> <p>This set is clearly an <a class="existingWikiWord" href="/nlab/show/ideal">ideal</a> in the <a class="existingWikiWord" href="/nlab/show/polynomial">polynomial</a> <a class="existingWikiWord" href="/nlab/show/ring">ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">k[X_1, \cdots, X_n]</annotation></semantics></math>, called the <em><a class="existingWikiWord" href="/nlab/show/vanishing+ideal">vanishing ideal</a></em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V(\mathcal{F})</annotation></semantics></math>.</p> </div> <div class="num_prop" id="PrimeVanishingIdealOfIrreducibleZariskiClosed"> <h6 id="proposition_3">Proposition</h6> <p>With <a class="existingWikiWord" href="/nlab/show/excluded+middle">excluded middle</a> then:</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/field">field</a>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo>⊂</mo><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">V(\mathcal{F}) \subset k^n</annotation></semantics></math> be a Zariski closed subset (def. <a class="maruku-ref" href="#ZariskiOpenSubsetsOnAffineSpace"></a>). Then the following are equivalent:</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V(\mathcal{F})</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/irreducible+closed+subset">irreducible closed subset</a>;</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/vanishing+ideal">vanishing ideal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">I(V(\mathcal{F}))</annotation></semantics></math> (def. <a class="maruku-ref" href="#VanishingIdeal"></a>) is a <a class="existingWikiWord" href="/nlab/show/prime+ideal">prime ideal</a>.</p> </li> </ol> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>In one direction, assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V(\mathcal{F})</annotation></semantics></math> is irreducible and consider <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo>∈</mo><mi>k</mi><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">f,g \in k[X_1, \cdots, X_n]</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>⋅</mo><mi>g</mi><mo>∈</mo><mi>I</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f \cdot g \in I(V(\mathcal{F}))</annotation></semantics></math>. We need to show that then already <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><mi>I</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f \in I(V(\mathcal{F}))</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∈</mo><mi>I</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g \in I(V(\mathcal{F}))</annotation></semantics></math>.</p> <p>Now since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> is a field, we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>⋅</mo><mi>g</mi><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>)</mo></mrow><mo>⇒</mo><mrow><mo>(</mo><mrow><mo>(</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mtext>or</mtext><mspace width="thinmathspace"></mspace><mi>g</mi><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( f(a_1, \cdots a_n) \cdot g(a_1, \cdots, a_n) = 0 \right) \Rightarrow \left( \left( f(a_1, \cdots, a_n) = 0 \,\text{or}\, g(a_1, \cdots, a_n) = 0 \right) \right) \,. </annotation></semantics></math></div> <p>This implies that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>V</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><mi>f</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>∪</mo><mi>V</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><mi>g</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> V(\mathcal{F}) \subset V(\{f\}) \cup V(\{g\}) </annotation></semantics></math></div> <p>and hence that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo>∩</mo><mi>F</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><mi>f</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mo>∪</mo><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo>∩</mo><mi>F</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><mi>g</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> V(\mathcal{F}) = (V(\mathcal{F}) \cap F(\{f\})) \,\,\cup\,\, (V(\mathcal{F}) \cap F(\{g\}) ) \,. </annotation></semantics></math></div> <p>But since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><mi>f</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V(\{f\})</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><mi>g</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V(\{g\})</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V(\mathcal{F})</annotation></semantics></math> are all closed, by construction, their intersections are closed and hence this is a decomposition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V(\mathcal{F})</annotation></semantics></math> as a union of closed subsets. Therefore now the assumption that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V(\mathcal{F})</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/irreducible+closed+subset">irreducible</a> implies that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd></mtd> <mtd><mrow><mo>(</mo><mspace width="thinmathspace"></mspace><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo>∩</mo><mi>V</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><mi>f</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mtext>or</mtext><mspace width="thinmathspace"></mspace><mrow><mo>(</mo><mspace width="thinmathspace"></mspace><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo>∩</mo><mi>V</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><mi>g</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>)</mo></mrow></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thickmathspace"></mspace></mtd> <mtd><mrow><mo>(</mo><mrow><mo>(</mo><mspace width="thinmathspace"></mspace><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>V</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><mi>f</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mtext>or</mtext><mspace width="thinmathspace"></mspace><mrow><mo>(</mo><mspace width="thinmathspace"></mspace><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>V</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><mi>g</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>)</mo></mrow><mo>)</mo></mrow></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thinmathspace"></mspace></mtd> <mtd><mrow><mo>(</mo><mrow><mo>(</mo><mspace width="thinmathspace"></mspace><mi>f</mi><mo>∈</mo><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mtext>or</mtext><mspace width="thinmathspace"></mspace><mrow><mo>(</mo><mspace width="thinmathspace"></mspace><mi>g</mi><mo>∈</mo><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>)</mo></mrow><mo>)</mo></mrow></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} & \left( \, V(\mathcal{F}) = V(\mathcal{F}) \cap V(\{f\}) \, \right) \,\text{or}\, \left( \, V(\mathcal{F}) = V(\mathcal{F}) \cap V(\{g\}) \, \right) \\ \Leftrightarrow \; & \left( \left( \, V(\mathcal{F}) \subset V(\{f\}) \, \right) \,\text{or}\, \left( \, V(\mathcal{F}) \subset V(\{g\}) \, \right) \right) \\ \Leftrightarrow \, & \left( \left( \, f \in I(X) \, \right) \,\text{or}\, \left( \, g \in I(X) \, \right) \right) \end{aligned} \,. </annotation></semantics></math></div> <p>Now for the converse, assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">I(V(\mathcal{F}))</annotation></semantics></math> is a prime ideal, and that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>∪</mo><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V(\mathcal{F}) = V(\mathcal{F}_1) \cup V(\mathcal{F}_2)</annotation></semantics></math>. We need to show that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V(\mathcal{F}) = V(\mathcal{F}_1)</annotation></semantics></math> or that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V(\mathcal{F}) = V(\mathcal{F}_2)</annotation></semantics></math>.</p> <p>Assume on the contrary, that there existed elements</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>\</mo><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mtext>and</mtext><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>b</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>\</mo><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex"> (a_1, \cdots, a_n) \in V(\mathcal{F}_1) \backslash V(\mathcal{F}_2) \;\text{and}\; (b_1, \cdots, b_n) \in V(\mathcal{F}_2) \backslash V(\mathcal{F}_1) \, </annotation></semantics></math></div> <p>Then in particular the vanishing ideals would not contain each other</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>¬</mo><mrow><mo>(</mo><mi>I</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⊂</mo><mi>I</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mtext>and</mtext><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mo>¬</mo><mrow><mo>(</mo><mi>I</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⊂</mo><mi>I</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \not\left( I(V(\mathcal{F}_1)) \subset I(V(\mathcal{F}_2)) \right) \,\,\,\text{and}\,\,\, \not\left( I(V(\mathcal{F}_2)) \subset I(V(\mathcal{F}_1)) \right) </annotation></semantics></math></div> <p>and hence there were polynomials</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><mi>I</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>\</mo><mi>I</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mtext>and</mtext><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mi>g</mi><mo>∈</mo><mi>I</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>\</mo><mi>I</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f\in I(V(\mathcal{F}_1)) \backslash I(V(\mathcal{F}_2)) \,\,\,\text{and}\,\,\, g \in I(V(\mathcal{F}_2)) \backslash I(V(\mathcal{F}_1)) \,. </annotation></semantics></math></div> <p>But since a product of polynomials vanishes at some point once one of the factors vanishes at that point, it would follows that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>⋅</mo><mi>g</mi><mo>∈</mo><mi>I</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∩</mo><mi>I</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>I</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> f \cdot g \in I(V(\mathcal{F}_1)) \cap I(V(\mathcal{F}_2)) = I(V(\mathcal{F})) \,, </annotation></semantics></math></div> <p>which were in contradiction to the assumption that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">I(V(\mathcal{F}))</annotation></semantics></math> is a prime ideal. Hence we have a <a class="existingWikiWord" href="/nlab/show/proof+by+contradiction">proof by contradiction</a>.</p> </div> <p>Proposition <a class="maruku-ref" href="#PrimeVanishingIdealOfIrreducibleZariskiClosed"></a> gives an <a class="existingWikiWord" href="/nlab/show/injection">injection</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mtext>irreducible Zariski closed</mtext></mtd></mtr> <mtr><mtd><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo>⊂</mo><msup><mi>k</mi> <mi>n</mi></msup></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mover><mo>↪</mo><mphantom><mi>AAA</mi></mphantom></mover><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mtext>prime ideals</mtext></mtd></mtr> <mtr><mtd><mi>I</mi><mo>∈</mo><mi>k</mi><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left\{ \array{ \text{irreducible Zariski closed} \\ V(\mathcal{F}) \subset k^n } \right\} \overset{\phantom{AAA}}{\hookrightarrow} \left\{ \array{ \text{prime ideals} \\ I \in k[X_1, \cdots, X_n] } \right\} \,. </annotation></semantics></math></div> <p>The following says that for <a class="existingWikiWord" href="/nlab/show/algebraically+closed+fields">algebraically closed fields</a> then this is in fact a <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a>:</p> <div class="num_prop" id="PrimeIdealClosedsubspaceBijection"> <h6 id="proposition_4">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo><mover><mi>k</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">k = \overline{k}</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/algebraically+closed+field">algebraically closed field</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>. Then the function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>IrrClSub</mi><mo stretchy="false">(</mo><msubsup><mi>𝔸</mi> <mi>k</mi> <mi>n</mi></msubsup><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mi>PrimeIdl</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>↦</mo><mphantom><mi>AAA</mi></mphantom></mover></mtd> <mtd><mi>I</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ IrrClSub(\mathbb{A}^n_k) &\overset{}{\longrightarrow}& PrimeIdl(k[X_1, \cdots, X_n]) \\ V(\mathcal{F}) &\overset{\phantom{AAA}}{\mapsto}& I(V(\mathcal{F})) } </annotation></semantics></math></div> <p>from prop. <a class="maruku-ref" href="#PrimeVanishingIdealOfIrreducibleZariskiClosed"></a> is a <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a>.</p> </div> <p>The <strong>proof</strong> uses <a class="existingWikiWord" href="/nlab/show/Hilbert%27s+Nullstellensatz">Hilbert's Nullstellensatz</a>.</p> <div class="num_remark"> <h6 id="remark">Remark</h6> <p><strong>(generalization to affine varieties)</strong></p> <p>Prop <a class="maruku-ref" href="#ZariskiClosedSubsetsInSpecR"></a> suggests to consider the set of <a class="existingWikiWord" href="/nlab/show/prime+ideals">prime ideals</a> of a <a class="existingWikiWord" href="/nlab/show/polynomial+ring">polynomial ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">k[X_1, \cdots, X_n]</annotation></semantics></math> for general <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> as more fundamental, in some sense, than the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">k^n</annotation></semantics></math>. Morover, the set of prime ideals makes sense for every <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>, not just <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>=</mo><mi>k</mi><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">R = k[X_1, \cdots, X_n]</annotation></semantics></math>, and hence this suggests to consider a Zariski topology on sets of prime ideals. This leads to the more general concept of Zariski topologies for <a class="existingWikiWord" href="/nlab/show/affine+varieties">affine varieties</a>, def. <a class="maruku-ref" href="#ZariskiClosedSubsetsInSpecR"></a> below.</p> </div> <h3 id="examples">Examples</h3> <div class="num_example" id="AffineSpaceOverInfiniteFieldNotHausdorff"> <h6 id="example">Example</h6> <p>If the <a class="existingWikiWord" href="/nlab/show/field">field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> is <em>not</em> a <a class="existingWikiWord" href="/nlab/show/finite+field">finite field</a>, then the Zariski topology on the <a class="existingWikiWord" href="/nlab/show/affine+space">affine space</a> (def. <a class="maruku-ref" href="#ZariskiOpenSubsetsOnAffineSpace"></a>) is <em>not</em> <a class="existingWikiWord" href="/nlab/show/Hausdorff+topological+space">Hausdorff</a>.</p> <p>This is because the solution set to a system of <a class="existingWikiWord" href="/nlab/show/polynomials">polynomials</a> over an infinite polynomial is always a <a class="existingWikiWord" href="/nlab/show/finite+set">finite set</a>. This means that in this case all the Zariski closed subsets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V(\mathcal{F})</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/finite+sets">finite sets</a>. This in turn implies that the <a class="existingWikiWord" href="/nlab/show/intersection">intersection</a> of <em>every</em> pair of <a class="existingWikiWord" href="/nlab/show/inhabited+set">non-empty</a> Zariski open subsets is <a class="existingWikiWord" href="/nlab/show/inhabited">non-empty</a>.</p> <p>But the Zariski topology is always <a class="existingWikiWord" href="/nlab/show/sober+topological+space">sober</a>, see prop. <a class="maruku-ref" href="#ZariskiTopologyIsSober"></a> below.</p> </div> <h2 id="OnAffineVarieties">On affine varieties</h2> <h3 id="definition_4">Definition</h3> <div class="num_defn" id="ZariskiClosedSubsetsInSpecR"> <h6 id="definition_5">Definition</h6> <p><strong>(Zariski topology on set of prime ideals)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PrimeIdl</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PrimeIdl(R)</annotation></semantics></math> for its set of <a class="existingWikiWord" href="/nlab/show/prime+ideals">prime ideals</a>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℱ</mi><mo>⊂</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">\mathcal{F} \subset R</annotation></semantics></math> any subset of elements of the ring, consider the subsets of those prime ideals that contain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℱ</mi></mrow><annotation encoding="application/x-tex">\mathcal{F}</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mrow><mo>{</mo><mi>p</mi><mo>∈</mo><mi>PrimeIdl</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thinmathspace"></mspace><mi>ℱ</mi><mo>⊂</mo><mi>p</mi><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> V(\mathcal{F}) \;\coloneqq\; \left\{ p \in PrimeIdl(R) \,\vert\, \mathcal{F} \subset p \right\} \,. </annotation></semantics></math></div> <p>These are called the <em>Zariski <a class="existingWikiWord" href="/nlab/show/closed+subsets">closed subsets</a></em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PrimeIdl</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PrimeIdl(R)</annotation></semantics></math>. Their <a class="existingWikiWord" href="/nlab/show/complements">complements</a> are called the <em>Zariski open subsets</em>.</p> </div> <div class="num_prop" id="WellDefinedZariskiTopologyOnSpecR"> <h6 id="proposition_5">Proposition</h6> <p><strong>(Zariski topology well defined)</strong></p> <p>Assuming <a class="existingWikiWord" href="/nlab/show/excluded+middle">excluded middle</a>, then:</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a>. Then the collection of Zariski open subsets (def. <a class="maruku-ref" href="#ZariskiClosedSubsetsInSpecR"></a>) in its set of <a class="existingWikiWord" href="/nlab/show/prime+ideals">prime ideals</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow></msub><mo>⊂</mo><mi>P</mi><mo stretchy="false">(</mo><mi>PrimeIdl</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \tau_{Spec(R)} \subset P(PrimeIdl(R)) </annotation></semantics></math></div> <p>satisfies the axioms of a <a class="existingWikiWord" href="/nlab/show/topological+space">topology</a>, the <em>Zariski topology</em>.</p> <p>This <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">(</mo><mi>PrimeIdl</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>τ</mi> <mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Spec(R) \coloneqq (PrimeIdl(R), \tau_{Spec(R)}) </annotation></semantics></math></div> <p>is called (the space underlying) the <em><a class="existingWikiWord" href="/nlab/show/prime+spectrum+of+a+commutative+ring">prime spectrum of the commutative ring</a></em>.</p> </div> <div class="proof" id="WellDefinedZariskiTopologyOnSpecRProof"> <h6 id="proof_4">Proof</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℱ</mi><mo>⊂</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">\mathcal{F} \subset R</annotation></semantics></math> write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℱ</mi><mo>⊂</mo><mi>I</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{F} \subset I(\mathcal{F})</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/ideal">ideal</a> which is generated by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℱ</mi></mrow><annotation encoding="application/x-tex">\mathcal{F}</annotation></semantics></math>. Evidently the Zariski closed subsets depend only on this ideal</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> V(I(\mathcal{F})) = V(\mathcal{F}) </annotation></semantics></math></div> <p>and therefore it is sufficient to consider the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V(\mathcal{F})</annotation></semantics></math> for the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℱ</mi><mo>⊂</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">\mathcal{F} \subset R</annotation></semantics></math> is not just a subset, but an ideal.</p> <p>So let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>F</mi> <mi>i</mi></msub><mo>∈</mo><mi>Idl</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{F_i \in Idl(R)\}_{i \in I}</annotation></semantics></math> be a set of ideals in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>⊂</mo><mi>PrimeIdl</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{V(\mathcal{F}_i) \subset PrimeIdl(R)\}_{i \in I}</annotation></semantics></math> be the corresponding set of Zariski closed subsets. We need to show that there exists <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℱ</mi> <mo>∪</mo></msub><mo>,</mo><msub><mi>ℱ</mi> <mo>∩</mo></msub><mo>⊂</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">\mathcal{F}_\cup, \mathcal{F}_\cap \subset R</annotation></semantics></math> such that</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mo>∩</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mo>∪</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\underset{i \in I}{\cap} V(\mathcal{F}_i) = V(\mathcal{F}_\cup)</annotation></semantics></math>;</p> </li> <li> <p>if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/finite+set">finite set</a> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mo>∪</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mo>∩</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\underset{i \in I}{\cup} V(\mathcal{F}_i) = V(\mathcal{F}_\cap)</annotation></semantics></math>.</p> </li> </ol> <p>We claim that</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℱ</mi> <mo>∪</mo></msub><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>ℱ</mi> <mi>i</mi></msub><mo>≔</mo><mrow><mo>{</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>f</mi> <mi>i</mi></msub><mspace width="thinmathspace"></mspace><mo>,</mo><mo>∈</mo><mi>R</mi><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><msub><mi>f</mi> <mi>i</mi></msub><mo>∈</mo><msub><mi>ℱ</mi> <mi>i</mi></msub><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex">\mathcal{F}_{\cup} = \underset{i \in I}{\sum} \mathcal{F}_i \coloneqq \left\{ \underset{i \in I}{\sum} f_i \,, \in R\;\vert\; f_i \in \mathcal{F}_i \right\}</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℱ</mi> <mo>∩</mo></msub><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>ℱ</mi> <mi>i</mi></msub><mo>≔</mo><mrow><mo>{</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>f</mi> <mi>i</mi></msub><mspace width="thinmathspace"></mspace><mo>∈</mo><mi>R</mi><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><msub><mi>f</mi> <mi>i</mi></msub><mo>∈</mo><msub><mi>ℱ</mi> <mi>i</mi></msub><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex">\mathcal{F}_{\cap} = \underset{i \in I}{\prod} \mathcal{F}_i \coloneqq \left\{ \underset{i \in I}{\prod} f_i \, \in R \;\vert\; f_i \in \mathcal{F}_i \right\}</annotation></semantics></math>,</p> </li> </ul> <p>Regarding the first point:</p> <p>By using the various definitions, we get the following chain of logical equivalences:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd></mtd> <mtd><mi>p</mi><mo>∈</mo><munder><mo>∩</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thickmathspace"></mspace></mtd> <mtd><munder><mo>∀</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mrow><mo>(</mo><mi>p</mi><mo>∈</mo><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>)</mo></mrow></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thickmathspace"></mspace></mtd> <mtd><munder><mo>∀</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mrow><mo>(</mo><msub><mi>ℱ</mi> <mi>i</mi></msub><mo>⊂</mo><mi>p</mi><mo>)</mo></mrow></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thickmathspace"></mspace></mtd> <mtd><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>ℱ</mi> <mi>i</mi></msub><mo>)</mo></mrow><mo>⊂</mo><mi>p</mi></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thickmathspace"></mspace></mtd> <mtd><mi>p</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>ℱ</mi> <mi>i</mi></msub><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} & p \in \underset{i \in I}{\cap} V(\mathcal{F}_i) \\ \Leftrightarrow\; & \underset{i \in I}{\forall} \left( p \in V(\mathcal{F}_i) \right) \\ \Leftrightarrow\; & \underset{i \in I}{\forall} \left( \mathcal{F}_i \subset p \right) \\ \Leftrightarrow\; & \left(\underset{i \in I}{\sum} \mathcal{F}_i\right) \subset p \\ \Leftrightarrow\; & p \in V\left( \underset{i \in I}{\sum} \mathcal{F}_i \right) \,. \end{aligned} </annotation></semantics></math></div> <p>Regarding the second point, in one direction we have the immediate implication</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd></mtd> <mtd><mi>p</mi><mo>∈</mo><munder><mo>∪</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thickmathspace"></mspace></mtd> <mtd><munder><mo>∃</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mrow><mo>(</mo><msub><mi>ℱ</mi> <mi>i</mi></msub><mo>⊂</mo><mi>p</mi><mo>)</mo></mrow></mtd></mtr> <mtr><mtd><mo>⇒</mo><mspace width="thickmathspace"></mspace></mtd> <mtd><munder><mo>∀</mo><mrow><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>ℱ</mi> <mi>i</mi></msub></mrow></munder><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>f</mi> <mi>i</mi></msub><mo>∈</mo><mi>p</mi><mo>)</mo></mrow></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thickmathspace"></mspace></mtd> <mtd><mi>p</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>ℱ</mi> <mi>i</mi></msub><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} & p \in \underset{i \in I}{\cup} V(\mathcal{F}_i) \\ \Leftrightarrow\; & \underset{i \in I}{\exists} \left( \mathcal{F}_i \subset p \right) \\ \Rightarrow \; & \underset{(f_i) \in \underset{i \in I}{\prod} \mathcal{F}_i}{\forall} \left( \underset{i \in I}{\prod} f_i \in p \right) \\ \Leftrightarrow\; & p \in V\left( \underset{i \in I}{\prod} \mathcal{F}_i \right) \,. \end{aligned} </annotation></semantics></math></div> <p>For the converse direction we need to show that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mi>p</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>ℱ</mi> <mi>i</mi></msub><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>⇒</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mi>p</mi><mo>∈</mo><munder><mo>∪</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( p \in V\left( \underset{i \in I}{\prod} \mathcal{F}_i \right) \right) \;\Rightarrow\; \left( p \in \underset{i \in I}{\cup} V(\mathcal{F}_1) \right) \,. </annotation></semantics></math></div> <p>hence that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><munder><mo>∀</mo><mrow><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>ℱ</mi> <mi>i</mi></msub></mrow></munder><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>f</mi> <mi>i</mi></msub><mo>∈</mo><mi>p</mi><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>⇒</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><munder><mo>∃</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mrow><mo>(</mo><munder><mo>∀</mo><mrow><msub><mi>f</mi> <mi>i</mi></msub><mo>∈</mo><msub><mi>ℱ</mi> <mi>i</mi></msub></mrow></munder><mrow><mo>(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo>∈</mo><mi>p</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( \underset{(f_i) \in \underset{i \in I}{\prod} \mathcal{F}_i }{\forall} \left( \underset{i \in I}{\prod} f_i \in p \right) \right) \;\Rightarrow\; \left( \underset{i \in I}{\exists} \left( \underset{f_i \in \mathcal{F}_i}{\forall} \left( f_i \in p \right) \right) \right) \,. </annotation></semantics></math></div> <p>By <a class="existingWikiWord" href="/nlab/show/excluded+middle">excluded middle</a>, this is equivalent to its <a class="existingWikiWord" href="/nlab/show/contraposition">contraposition</a>, which by <a class="existingWikiWord" href="/nlab/show/de+Morgan%27s+law">de Morgan's law</a> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><munder><mo>∀</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mrow><mo>(</mo><munder><mo>∃</mo><mrow><msub><mi>f</mi> <mi>i</mi></msub><mo>∈</mo><msub><mi>ℱ</mi> <mi>i</mi></msub></mrow></munder><mo>¬</mo><mrow><mo>(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo>∈</mo><mi>p</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>⇒</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><munder><mo>∃</mo><mrow><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>ℱ</mi> <mi>i</mi></msub></mrow></munder><mo>¬</mo><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>f</mi> <mi>i</mi></msub><mo>∈</mo><mi>p</mi><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( \underset{i \in I}{\forall} \left( \underset{f_i \in \mathcal{F}_i}{\exists} \not \left( f_i \in p \right) \right) \right) \;\Rightarrow\; \left( \underset{(f_i) \in \underset{i \in I}{\prod} \mathcal{F}_i }{\exists} \not \left( \underset{i \in I}{\prod} f_i \in p \right) \right) \,. </annotation></semantics></math></div> <p>This holds by the assumption that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/prime+ideal">prime ideal</a>.</p> </div> <h3 id="PropertiesOnAffineVarieties">Properties</h3> <p>We discuss some properties of the Zariski topology on <a class="existingWikiWord" href="/nlab/show/prime+spectra+of+commutative+rings">prime spectra of commutative rings</a>.</p> <h4 id="topological_closures_2">Topological closures</h4> <div class="num_lemma" id="ZariskiClorsuredOfPont"> <h6 id="lemma">Lemma</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/topological+closure">topological closure</a> of points)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> and consider <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>PrimeIdl</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>τ</mi> <mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(R) = (PrimeIdl(R), \tau_{Spec(R)})</annotation></semantics></math> its <a class="existingWikiWord" href="/nlab/show/prime+spectrum+of+a+commutative+ring">prime spectrum</a> equipped with the Zariski topology (def. <a class="maruku-ref" href="#ZariskiClosedSubsetsInSpecR"></a>).</p> <p>Then the <a class="existingWikiWord" href="/nlab/show/topological+closure">topological closure</a> of a point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∈</mo><mi>PrimeIdl</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p \in PrimeIdl(R)</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>PrimeIdl</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V(p) \subset PrimeIdl(R)</annotation></semantics></math> (def. <a class="maruku-ref" href="#ZariskiClosedSubsetsInSpecR"></a>).</p> </div> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>By definition the topological closure of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>p</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{p\}</annotation></semantics></math> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Cl</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><mi>p</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>=</mo><munder><mo>∩</mo><mfrac linethickness="0"><mrow><mrow><mi>I</mi><mo>∈</mo><mi>Idl</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mi>p</mi><mo>∈</mo><mi>V</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></mrow></mrow></mfrac></munder><mi>V</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Cl(\{p\})=\underset{ {I \in Idl(R) } \atop { p \in V(I) } }{\cap} V(I) \,. </annotation></semantics></math></div> <p>Hence unwinding the definitions, we have the following sequence of logical equivalences:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd></mtd> <mtd><mi>q</mi><mo>∈</mo><mi>Cl</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><mi>q</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thickmathspace"></mspace></mtd> <mtd><mi>q</mi><mo>∈</mo><munder><mo>∩</mo><mfrac linethickness="0"><mrow><mrow><mi>I</mi><mo>∈</mo><mi>Idl</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mi>p</mi><mo>∈</mo><mi>V</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></mrow></mrow></mfrac></munder><mi>V</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thickmathspace"></mspace></mtd> <mtd><munder><mo>∀</mo><mfrac linethickness="0"><mrow><mrow><mi>I</mi><mo>∈</mo><mi>Idl</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mi>I</mi><mo>⊂</mo><mi>p</mi></mrow></mrow></mfrac></munder><mo stretchy="false">(</mo><mi>q</mi><mo>∈</mo><mi>V</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thickmathspace"></mspace></mtd> <mtd><munder><mo>∀</mo><mfrac linethickness="0"><mrow><mrow><mi>I</mi><mo>∈</mo><mi>Idl</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mi>I</mi><mo>⊂</mo><mi>p</mi></mrow></mrow></mfrac></munder><mo stretchy="false">(</mo><mi>I</mi><mo>⊂</mo><mi>q</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thickmathspace"></mspace></mtd> <mtd><mi>p</mi><mo>⊂</mo><mi>q</mi></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thickmathspace"></mspace></mtd> <mtd><mi>q</mi><mo>∈</mo><mi>V</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} & q \in Cl(\{q\}) \\ \Leftrightarrow\; & q \in \underset{ {I \in Idl(R)} \atop { p \in V(I) } }{\cap} V(I) \\ \Leftrightarrow\; & \underset{ { I \in Idl(R) } \atop { I \subset p } }{\forall} (q \in V(I)) \\ \Leftrightarrow\; & \underset{ { I \in Idl(R) } \atop { I \subset p } }{\forall} (I \subset q) \\ \Leftrightarrow\; & p \subset q \\ \Leftrightarrow\; & q \in V(p) \end{aligned} </annotation></semantics></math></div></div> <p>Recall:</p> <div class="num_lemma" id="PrimeIdealTheorem"> <h6 id="lemma_2">Lemma</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/prime+ideal+theorem">prime ideal theorem</a>)</strong></p> <p>Assuming the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a> or at least the <a class="existingWikiWord" href="/nlab/show/ultrafilter+principle">ultrafilter principle</a> then:</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>⊂</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">I \subset R</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/proper+ideal">proper ideal</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is contained in some <a class="existingWikiWord" href="/nlab/show/prime+ideal">prime ideal</a>.</p> </div> <p>The <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a> even implies that every proper ideal is contained in a <a class="existingWikiWord" href="/nlab/show/maximal+ideal">maximal ideal</a> (by <a href="maximal+ideal#EveryProperIdealisContainedInAMaximalOne">this prop.</a>).</p> <div class="num_prop" id="MaximalIdealsAreClosedPoints"> <h6 id="proposition_6">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/maximal+ideals">maximal ideals</a> are <a class="existingWikiWord" href="/nlab/show/closed+points">closed points</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a>, consider the <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>PrimeIdl</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>τ</mi> <mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(R) = (PrimeIdl(R),\tau_{Spec(R)})</annotation></semantics></math>, i.e. its <a class="existingWikiWord" href="/nlab/show/prime+spectrum+of+a+commutative+ring">prime spectrum</a> equipped with the Zariski topology from def. <a class="maruku-ref" href="#ZariskiClosedSubsetsInSpecR"></a>.</p> <p>Then the <a class="existingWikiWord" href="/nlab/show/maximal+ideals">maximal ideals</a> inside the prime ideals constitute <a class="existingWikiWord" href="/nlab/show/closed+points">closed points</a>.</p> <p>Assuming the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a> or at least the <a class="existingWikiWord" href="/nlab/show/ultrafilter+principle">ultrafilter principle</a> then also the converse is true:</p> <p>Then the inclusion of <a class="existingWikiWord" href="/nlab/show/maximal+ideals">maximal ideals</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔪</mi><mo>∈</mo><mi>MaxIdl</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>PrimeIdl</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{m} \in MaxIdl(R) \subset PrimeIdl(R)</annotation></semantics></math> into all <a class="existingWikiWord" href="/nlab/show/prime+ideals">prime ideals</a> is precisely the inclusion of the subset of <a class="existingWikiWord" href="/nlab/show/closed+points">closed points</a> into all points of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(R)</annotation></semantics></math>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ClosedPoints</mi><mo stretchy="false">(</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>MaxIdl</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>PrimeIdl</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> ClosedPoints(Spec(R)) \simeq MaxIdl(R) \subset PrimeIdl(R) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_6">Proof</h6> <p>By lemma <a class="maruku-ref" href="#ZariskiClorsuredOfPont"></a> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Cl</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><mi>p</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>=</mo><mi>V</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Cl(\{p\}) = V(p) </annotation></semantics></math></div> <p>and hence we need to show that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>𝔪</mi><mo stretchy="false">}</mo><mo>=</mo><mi>V</mi><mo stretchy="false">(</mo><mi>𝔪</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \{\mathfrak{m}\} = V(\mathfrak{m}) </annotation></semantics></math></div> <p>precisely if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔪</mi></mrow><annotation encoding="application/x-tex">\mathfrak{m}</annotation></semantics></math> is maximal.</p> <p>In one direction, assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔪</mi></mrow><annotation encoding="application/x-tex">\mathfrak{m}</annotation></semantics></math> is maximal. By definition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>𝔪</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V(\mathfrak{m})</annotation></semantics></math> contains all the prime ideals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔪</mi><mo>⊂</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">\mathfrak{m} \subset p</annotation></semantics></math>. That <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔪</mi></mrow><annotation encoding="application/x-tex">\mathfrak{m}</annotation></semantics></math> is maximal means that it is not contained in a larger proper ideal, in particular not in any larger prime ideal, and hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>𝔪</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mi>𝔪</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">V(\mathfrak{m}) = \{\mathfrak{m}\}</annotation></semantics></math>.</p> <p>In the other direction, assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔪</mi></mrow><annotation encoding="application/x-tex">\mathfrak{m}</annotation></semantics></math> is a prime ideal such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>𝔪</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mi>𝔪</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">V(\mathfrak{m}) = \{\mathfrak{m}\}</annotation></semantics></math>. By definition this means equivalently that the only prime ideal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔪</mi><mo>⊂</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">\mathfrak{m} \subset p</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔪</mi></mrow><annotation encoding="application/x-tex">\mathfrak{m}</annotation></semantics></math> itself. We need to show that more generally <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔪</mi><mo>⊂</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">\mathfrak{m} \subset I</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/proper+ideal">proper ideal</a> implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔪</mi><mo>=</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">\mathfrak{m} = I</annotation></semantics></math>.</p> <p>But the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a>/<a class="existingWikiWord" href="/nlab/show/ultrafilter+principle">ultrafilter principle</a> imply the <a class="existingWikiWord" href="/nlab/show/prime+ideal+theorem">prime ideal theorem</a> (lemma <a class="maruku-ref" href="#PrimeIdealTheorem"></a>), which says that there is a prime ideal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>⊂</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">I \subset p</annotation></semantics></math>, hence a sequence of inclusions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔪</mi><mo>⊂</mo><mi>I</mi><mo>⊂</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">\mathfrak{m} \subset I \subset p</annotation></semantics></math>. Since this implies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔪</mi><mo>⊂</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">\mathfrak{m} \subset p</annotation></semantics></math>, we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔪</mi><mo>=</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">\mathfrak{m} = p</annotation></semantics></math>, hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>=</mo><mi>𝔪</mi></mrow><annotation encoding="application/x-tex">I = \mathfrak{m}</annotation></semantics></math>.</p> </div> <h4 id="irreducible_closed_subsets_as_prime_ideals_2">Irreducible closed subsets as prime ideals</h4> <div class="num_prop" id="ZariskiIrreducibleClosedSubsetsArePreciselyPrimeIdeals"> <h6 id="proposition_7">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/irreducible+closed+subsets">irreducible closed subsets</a> correspond to <a class="existingWikiWord" href="/nlab/show/prime+ideals">prime ideals</a>)</strong></p> <p>With <a class="existingWikiWord" href="/nlab/show/excluded+middle">excluded middle</a> then:</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℱ</mi><mo>⊂</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">\mathcal{F} \subset R</annotation></semantics></math> be an ideal in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>, hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V(\mathcal{F}) \subset Spec(R)</annotation></semantics></math> is a Zariski closed subset in the <a class="existingWikiWord" href="/nlab/show/prime+spectrum+of+a+commutative+ring">prime spectrum</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>. Then the following are equivalent:</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℱ</mi></mrow><annotation encoding="application/x-tex">\mathcal{F}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/radical+ideal">radical ideal</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V(\mathcal{F})</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/irreducible+closed+subset">irreducible closed subset</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℱ</mi></mrow><annotation encoding="application/x-tex">\mathcal{F}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/prime+ideal">prime ideal</a>.</p> </li> </ol> </div> <div class="proof"> <h6 id="proof_7">Proof</h6> <p>In one direction, assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V(\mathcal{F})</annotation></semantics></math> is irreducible, and that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">f,g \in R</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>⋅</mo><mi>g</mi><mo>∈</mo><mi>ℱ</mi></mrow><annotation encoding="application/x-tex">f \cdot g \in \mathcal{F}</annotation></semantics></math>. We need to show that then already <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><mi>ℱ</mi></mrow><annotation encoding="application/x-tex">f \in \mathcal{F}</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∈</mo><mi>ℱ</mi></mrow><annotation encoding="application/x-tex">g \in \mathcal{F}</annotation></semantics></math>.</p> <p>To this end, first observe that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>V</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∪</mo><mi>V</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> V(\mathcal{F}) \subset V((f)) \cup V((g)) \,. </annotation></semantics></math></div> <p>This is because</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd></mtd> <mtd><mi>p</mi><mo>∈</mo><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thickmathspace"></mspace></mtd> <mtd><mi>ℱ</mi><mo>⊂</mo><mi>p</mi></mtd></mtr> <mtr><mtd><mo>⇒</mo><mspace width="thickmathspace"></mspace></mtd> <mtd><mi>f</mi><mo>⋅</mo><mi>g</mi><mo>∈</mo><mi>p</mi></mtd></mtr> <mtr><mtd><mo>⇒</mo><mspace width="thickmathspace"></mspace></mtd> <mtd><mrow><mo>(</mo><mi>f</mi><mo>∈</mo><mi>p</mi><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mtext>or</mtext><mspace width="thinmathspace"></mspace><mrow><mo>(</mo><mi>g</mi><mo>∈</mo><mi>p</mi><mo>)</mo></mrow></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thickmathspace"></mspace></mtd> <mtd><mrow><mo>(</mo><mi>p</mi><mo>∈</mo><mi>V</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mtext>or</mtext><mspace width="thinmathspace"></mspace><mrow><mo>(</mo><mi>p</mi><mo>∈</mo><mi>V</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>)</mo></mrow></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thickmathspace"></mspace></mtd> <mtd><mi>p</mi><mo>∈</mo><mi>V</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>∪</mo><mi>V</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} & p \in V(\mathcal{F}) \\ \Leftrightarrow\; & \mathcal{F} \subset p \\ \Rightarrow \; & f \cdot g \in p \\ \Rightarrow\; & \left( f \in p \right) \,\text{or}\, \left( g \in p \right) \\ \Leftrightarrow\; & \left( p \in V(g) \right) \,\text{or}\, \left( p \in V(f) \right) \\ \Leftrightarrow\; & p \in V(f) \cup V(g) \,, \end{aligned} </annotation></semantics></math></div> <p>where the implication in the middle uses that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is a prime ideal.</p> <p>It follows that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mi>V</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>∩</mo><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>∪</mo><mrow><mo>(</mo><mi>V</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>∩</mo><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> V(\mathcal{F}) \;=\; \left( V(f) \cap V(\mathcal{F}) \right) \cup \left( V(g) \cap V(\mathcal{F}) \right) \,. </annotation></semantics></math></div> <p>This is a decomposition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V(\mathcal{F})</annotation></semantics></math> as a union of closed subsets, hence the assumption that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V(\mathcal{F})</annotation></semantics></math> is irreducible implies that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd></mtd> <mtd><mrow><mo>(</mo><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>V</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>∩</mo><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mtext>or</mtext><mspace width="thinmathspace"></mspace><mrow><mo>(</mo><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>V</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>∩</mo><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo>)</mo></mrow></mtd></mtr> <mtr><mtd><mo>⇔</mo></mtd> <mtd><mrow><mo>(</mo><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>V</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mtext>or</mtext><mspace width="thinmathspace"></mspace><mrow><mo>(</mo><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>V</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>)</mo></mrow></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thinmathspace"></mspace></mtd> <mtd><mrow><mo>(</mo><mi>f</mi><mo>∈</mo><mi>nil</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mtext>or</mtext><mspace width="thinmathspace"></mspace><mrow><mo>(</mo><mi>g</mi><mo>∈</mo><mi>nil</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo>)</mo></mrow></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thinmathspace"></mspace></mtd> <mtd><mrow><mo>(</mo><mi>f</mi><mo>∈</mo><mi>ℱ</mi><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mtext>or</mtext><mspace width="thinmathspace"></mspace><mrow><mo>(</mo><mi>g</mi><mo>∈</mo><mi>ℱ</mi><mo>)</mo></mrow><mo>,</mo></mtd></mtr> <mtr><mtd></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} & \left( V(\mathcal{F}) = V(f) \cap V(\mathcal{F}) \right) \,\text{or}\, \left( V(\mathcal{F}) = V(g) \cap V(\mathcal{F}) \right) \\ \Leftrightarrow & \left( V(\mathcal{F}) \subset V(f) \right) \,\text{or}\, \left( V(\mathcal{F}) \subset V(g) \right)\\ \Leftrightarrow\,& \left( f \in nil(\mathcal{F}) \right) \,\text{or}\, \left( g \in nil(\mathcal{F}) \right)\\ \Leftrightarrow\,& \left( f \in \mathcal{F} \right) \,\text{or}\, \left( g \in \mathcal{F} \right),\\ \end{aligned} </annotation></semantics></math></div> <p>where the last equivalence uses the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℱ</mi></mrow><annotation encoding="application/x-tex">\mathcal{F}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/radical+ideal">radical ideal</a>.</p> <p>Now for the converse. Assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℱ</mi></mrow><annotation encoding="application/x-tex">\mathcal{F}</annotation></semantics></math> is a prime ideal and that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>∪</mo><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V(\mathcal{F}) = V(\mathcal{F}_1) \cup V(\mathcal{F}_2)</annotation></semantics></math>. Observe (as in the <a href="#WellDefinedZariskiTopologyOnSpecRProof">proof</a> of prop. <a class="maruku-ref" href="#WellDefinedZariskiTopologyOnSpecR"></a>) that this means equivalently that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℱ</mi><mo>=</mo><msub><mi>ℱ</mi> <mn>1</mn></msub><mo>⋅</mo><msub><mi>ℱ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{F} = \mathcal{F}_1 \cdot \mathcal{F}_2</annotation></semantics></math>. We need to show that then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V(\mathcal{F}) = V(\mathcal{F}_1)</annotation></semantics></math> or that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>ℱ</mi><mo>=</mo><mi>V</mi><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V(\mathcal{F} = V(\mathcal{F}_2))</annotation></semantics></math>.</p> <p>Suppose on the contrary that neither <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℱ</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{F}_1</annotation></semantics></math> nor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℱ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{F}_2</annotation></semantics></math> coincided with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℱ</mi></mrow><annotation encoding="application/x-tex">\mathcal{F}</annotation></semantics></math>. This means that there were elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><msub><mi>ℱ</mi> <mn>1</mn></msub><mo>\</mo><mi>ℱ</mi></mrow><annotation encoding="application/x-tex">f \in \mathcal{F}_1 \backslash \mathcal{F}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∈</mo><msub><mi>ℱ</mi> <mn>2</mn></msub><mo>\</mo><mi>ℱ</mi></mrow><annotation encoding="application/x-tex">g \in \mathcal{F}_2 \backslash \mathcal{F}</annotation></semantics></math> such that still <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>⋅</mo><mi>g</mi><mo>∈</mo><mi>ℱ</mi></mrow><annotation encoding="application/x-tex">f \cdot g \in \mathcal{F}</annotation></semantics></math>, in contradiction to the assumption. Hence we have a <a class="existingWikiWord" href="/nlab/show/proof+by+contradiction">proof by contradiction</a>.</p> </div> <p>As a corollary:</p> <div class="num_prop" id="ZariskiTopologyIsSober"> <h6 id="proposition_8">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/schemes+are+sober">Zariski topology on prime spectra is sober</a>)</strong></p> <p>With <a class="existingWikiWord" href="/nlab/show/excluded+middle">excluded middle</a> and <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a> (or at least the <a class="existingWikiWord" href="/nlab/show/ultrafilter+principle">ultrafilter principle</a>) then:</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a>. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(R)</annotation></semantics></math> (its <a class="existingWikiWord" href="/nlab/show/prime+spectrum+of+a+commutative+ring">prime spectr</a> equipped with the Zariski topology of def. <a class="maruku-ref" href="#ZariskiClosedSubsetsInSpecR"></a>) is a <a class="existingWikiWord" href="/nlab/show/sober+topological+space">sober topological space</a>.</p> </div> <div class="proof"> <h6 id="proof_8">Proof</h6> <p>We need to show that the function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Cl</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>PrimeIdl</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>IrrClSub</mi><mo stretchy="false">(</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Cl(\{-\}) \;\colon\; PrimeIdl(R) \longrightarrow IrrClSub(Spec(R)) </annotation></semantics></math></div> <p>which sends a point to its topological closure, is a <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a>.</p> <p>By lemma <a class="maruku-ref" href="#ZariskiClorsuredOfPont"></a> this function is given by sending a <a class="existingWikiWord" href="/nlab/show/prime+ideal">prime ideal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∈</mo><mi>PrimeIdl</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p \in PrimeIdl(R)</annotation></semantics></math> to the Zariski closed subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V(p)</annotation></semantics></math>. That this is a bijection is the statement of prop. <a class="maruku-ref" href="#ZariskiIrreducibleClosedSubsetsArePreciselyPrimeIdeals"></a>.</p> </div> <h3 id="examples_2">Examples</h3> <div class="num_example" id="AffinSpaceAsPrimeSpectrum"> <h6 id="example_2">Example</h6> <p><strong>(affine space as prime spectrum)</strong></p> <p>Reconsider the case where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>=</mo><mi>k</mi><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">R = k[X_1,\cdots, X_n]</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/polynomial+ring">polynomial ring</a>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/field">field</a>, as in the discussion of the naive affine space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">k^n</annotation></semantics></math> <a href="#OnAffineSpace">above</a>.</p> <p>Observe that, by <a class="maruku-ref" href="#MaximalIdealsAreClosedPoints"></a>, the <a class="existingWikiWord" href="/nlab/show/closed+points">closed points</a> in the <a class="existingWikiWord" href="/nlab/show/prime+spectrum+of+a+commutative+ring">prime spectrum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(k[X_1, \cdots, X_n])</annotation></semantics></math> correspond to the <a class="existingWikiWord" href="/nlab/show/maximal+ideals">maximal ideals</a> in the <a class="existingWikiWord" href="/nlab/show/polynomial+ring">polynomial ring</a>. These are of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>≔</mo><mrow><mo>(</mo><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>−</mo><msub><mi>a</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>2</mn></msub><mo>−</mo><msub><mi>a</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>⋯</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>n</mi></msub><mo>−</mo><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> (a_1, \cdots, a_n) \coloneqq \left( (X_1 - a_1) \cdot (X_2 - a_2) \cdots (X_n - a_n) \right) </annotation></semantics></math></div> <p>and hence are in bijection with the points of the naive affine space</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>k</mi> <mi>n</mi></msup><mo>≃</mo><mi>MaxIdl</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> k^n \simeq MaxIdl(k[X_1, \cdots, X_n]) \,. </annotation></semantics></math></div> <p>There is however also <a class="existingWikiWord" href="/nlab/show/prime+ideals">prime ideals</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">k[X_1, \cdots, X_n]</annotation></semantics></math> which are not maximal. In particular there is the 0-ideal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0)</annotation></semantics></math>.</p> </div> <div class="num_prop" id="SpecZ"> <h6 id="proposition_9">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Spec%28Z%29">Spec(Z)</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>=</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">R = \mathbb{Z}</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> of <a class="existingWikiWord" href="/nlab/show/integers">integers</a>. Consider the corresponding Zariski <a class="existingWikiWord" href="/nlab/show/prime+spectrum+of+a+commutative+ring">prime spectrum</a> (prop. <a class="maruku-ref" href="#WellDefinedZariskiTopologyOnSpecR"></a>) <a class="existingWikiWord" href="/nlab/show/Spec%28Z%29">Spec(Z)</a>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/prime+ideals">prime ideals</a> of the ring of integers are</p> <ol> <li> <p>the ideals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p)</annotation></semantics></math> generated by <a class="existingWikiWord" href="/nlab/show/prime+numbers">prime numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> (this special case is what motivates the terminology “prime ideal”);</p> </li> <li> <p>the ideal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">(0) = \{0\}</annotation></semantics></math>.</p> </li> </ol> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>PrimeIdl</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mspace width="thickmathspace"></mspace><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>7</mn><mo>,</mo><mn>11</mn><mo>,</mo><mi>⋯</mi><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> PrimeIdl(\mathbb{Z}) = \left\{ 0, \; 2, 3, 5, 7, 11, \cdots \right\} \,. </annotation></semantics></math></div> <p>All the prime ideals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">p \geq 2</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/maximal+ideals">maximal ideals</a>. Hence by prop. <a class="maruku-ref" href="#MaximalIdealsAreClosedPoints"></a> these are <a class="existingWikiWord" href="/nlab/show/closed+points">closed points</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(\mathbb{Z})</annotation></semantics></math>.</p> <p>Only the prime ideal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0)</annotation></semantics></math> is not maximal, hence the point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0)</annotation></semantics></math> is not closed. Its closure is the entire space</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Cl</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>=</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Cl(\{0\}) = Spec(\mathbb{Z}) \,. </annotation></semantics></math></div> <p>To see this, notice that in fact <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(\mathbb{Z})</annotation></semantics></math> is the only closed subset containing the point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0)</annotation></semantics></math>. This is because</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd></mtd> <mtd><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>∈</mo><mi>V</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thickmathspace"></mspace></mtd> <mtd><mi>I</mi><mo>⊂</mo><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thickmathspace"></mspace></mtd> <mtd><mi>I</mi><mo>=</mo><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} & (0) \in V(I) \\ \Leftrightarrow\; & I \subset (0) \\ \Leftrightarrow\; & I = (0) \end{aligned} </annotation></semantics></math></div> <p>and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V(0) = Spec(\mathbb{Z})</annotation></semantics></math>, because</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>∈</mo><mi>V</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⇔</mo><mo stretchy="false">(</mo><mn>0</mn><mo>⊂</mo><mi>p</mi><mo stretchy="false">)</mo><mo>⇔</mo><mi>true</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (p \in V(0)) \Leftrightarrow (0 \subset p) \Leftrightarrow true \,. </annotation></semantics></math></div></div> <h2 id="InTermsOfGaloisConnections">In terms of Galois connections</h2> <p>We now discuss how all of the above constructions and statements, and a bit more, follows immediately as a special case of the general theory of what is called <em><a class="existingWikiWord" href="/nlab/show/Galois+connections">Galois connections</a></em> or <em><a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a> between <a class="existingWikiWord" href="/nlab/show/posets">posets</a></em>.</p> <h3 id="BackgroundOnGaloisConnections">Background on Galois connections</h3> <div class="num_defn" id="GaloisConnection"> <h6 id="definition_6">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Galois+connection">Galois connection</a> induced from a <a class="existingWikiWord" href="/nlab/show/relation">relation</a>)</strong></p> <p>Consider two <a class="existingWikiWord" href="/nlab/show/sets">sets</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>∈</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">X,Y \in Set</annotation></semantics></math> and a <a class="existingWikiWord" href="/nlab/show/relation">relation</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>↪</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> E \hookrightarrow X \times Y \,. </annotation></semantics></math></div> <p>Define two <a class="existingWikiWord" href="/nlab/show/functions">functions</a> between their <a class="existingWikiWord" href="/nlab/show/power+sets">power sets</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(X), P(Y)</annotation></semantics></math>, as follows. (In the following we write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E(x, y)</annotation></semantics></math> to abbreviate the formula <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">(x, y) \in E</annotation></semantics></math>.)</p> <ol> <li> <p>Define</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>E</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> V_E \;\colon\; P(X) \longrightarrow P(Y) </annotation></semantics></math></div> <p>by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>≔</mo><mrow><mo>{</mo><mi>y</mi><mo>∈</mo><mi>Y</mi><mo stretchy="false">|</mo><munder><mo>∀</mo><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow></munder><mrow><mo>(</mo><mrow><mo>(</mo><mi>x</mi><mo>∈</mo><mi>S</mi><mo>)</mo></mrow><mo>⇒</mo><mi>E</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> V_E(S) \coloneqq \left\{ y \in Y \vert \underset{x \in X}{\forall} \left( \left(x \in S\right) \Rightarrow E(x, y) \right) \right\} </annotation></semantics></math></div></li> <li> <p>Define</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>E</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> I_E \;\colon\; P(Y) \longrightarrow P(X) </annotation></semantics></math></div> <p>by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>≔</mo><mrow><mo>{</mo><mi>x</mi><mo>∈</mo><mi>X</mi><mo stretchy="false">|</mo><munder><mo>∀</mo><mrow><mi>y</mi><mo>∈</mo><mi>Y</mi></mrow></munder><mrow><mo>(</mo><mrow><mo>(</mo><mi>y</mi><mo>∈</mo><mi>T</mi><mo>)</mo></mrow><mo>⇒</mo><mi>E</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> I_E(T) \coloneqq \left\{x \in X \vert \underset{y \in Y}{\forall} \left( \left(y \in T \right) \Rightarrow E(x, y) \right)\right\} </annotation></semantics></math></div></li> </ol> </div> <div class="num_prop" id="GaloisConnectionAsAdjunction"> <h6 id="proposition_10">Proposition</h6> <p>The construction in def. <a class="maruku-ref" href="#GaloisConnection"></a> has the following properties:</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>E</mi></msub></mrow><annotation encoding="application/x-tex">V_E</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>E</mi></msub></mrow><annotation encoding="application/x-tex">I_E</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/contravariant+functor">contravariant</a> order-preserving in that</p> <ol> <li> <p>if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>⊂</mo><mi>S</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">S \subset S'</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>S</mi><mo>′</mo><mo stretchy="false">)</mo><mo>⊂</mo><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V_E(S') \subset V_E(S)</annotation></semantics></math>;</p> </li> <li> <p>if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>⊂</mo><mi>T</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">T \subset T'</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mo>′</mo><mo stretchy="false">)</mo><mo>⊂</mo><msub><mi>I</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">I_E(T') \subset I_E(T)</annotation></semantics></math></p> </li> </ol> </li> <li> <p>The <em><a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> law</em> holds: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mi>T</mi><mo>⊂</mo><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo lspace="0em" rspace="thinmathspace">LeftRightarrow</mo><mspace width="thinmathspace"></mspace><mrow><mo>(</mo><mi>S</mi><mo>⊂</mo><msub><mi>I</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \left( T \subset V_E(S) \right) \,\LeftRightarrow\, \left( S \subset I_E(T) \right) </annotation></semantics></math></p> <p>which we denote by writing</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><munderover><mo>⊥</mo><munder><mo>⟶</mo><mrow><msub><mi>V</mi> <mi>E</mi></msub></mrow></munder><mover><mo>⟵</mo><mrow><msub><mi>I</mi> <mi>E</mi></msub></mrow></mover></munderover><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex"> P(X) \underoverset{\underset{V_E}{\longrightarrow}}{\overset{I_E}{\longleftarrow}}{\bot} P(Y)^{op} </annotation></semantics></math></div></li> <li> <p>both <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>E</mi></msub></mrow><annotation encoding="application/x-tex">V_E</annotation></semantics></math> as well as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>E</mi></msub></mrow><annotation encoding="application/x-tex">I_E</annotation></semantics></math> take <a class="existingWikiWord" href="/nlab/show/unions">unions</a> to <a class="existingWikiWord" href="/nlab/show/intersections">intersections</a>.</p> </li> </ol> </div> <div class="proof"> <h6 id="proof_9">Proof</h6> <p>Regarding the first point: the larger <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is, the more conditions that are placed on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> in order to belong to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V_E(S)</annotation></semantics></math>, and so the smaller <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V_E(S)</annotation></semantics></math> will be.</p> <p>Regarding the second point: This is because both these conditions are equivalent to the condition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>×</mo><mi>T</mi><mo>⊂</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">S \times T \subset E</annotation></semantics></math>.</p> <p>Regarding the third point: Observe that in a poset such as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(Y)</annotation></semantics></math>, we have that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A = B</annotation></semantics></math> iff for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>≤</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">C \leq A</annotation></semantics></math> iff <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>≤</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">C \leq B</annotation></semantics></math> (this is the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a> applied to posets). It follows that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>T</mi><mo>⊂</mo><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">⋃</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>S</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><mi>iff</mi></mtd> <mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">⋃</mo> <mrow><mi>i</mi><mo>:</mo><mi>I</mi></mrow></munder><msub><mi>S</mi> <mi>i</mi></msub><mo>⊂</mo><msub><mi>I</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mi>iff</mi></mtd> <mtd><msub><mo>∀</mo> <mrow><mi>i</mi><mo>:</mo><mi>I</mi></mrow></msub><msub><mi>S</mi> <mi>i</mi></msub><mo>⊂</mo><msub><mi>I</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mi>iff</mi></mtd> <mtd><msub><mo>∀</mo> <mrow><mi>i</mi><mo>:</mo><mi>I</mi></mrow></msub><mi>T</mi><mo>⊂</mo><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mi>iff</mi></mtd> <mtd><mi>T</mi><mo>⊂</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">⋂</mo> <mrow><mi>i</mi><mo>:</mo><mi>I</mi></mrow></munder><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ T \subset V_E(\bigcup_{i \in I} S_i) & iff & \bigcup_{i: I} S_i \subset I_E(T) \\ & iff & \forall_{i: I} S_i \subset I_E(T) \\ & iff & \forall_{i: I} T \subset V_E(S_i) \\ & iff & T \subset \bigcap_{i: I} V_E(S_i) } </annotation></semantics></math></div> <p>and we conclude <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">⋃</mo> <mrow><mi>i</mi><mo>:</mo><mi>I</mi></mrow></msub><msub><mi>S</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">⋂</mo> <mrow><mi>i</mi><mo>:</mo><mi>I</mi></mrow></msub><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V_E(\bigcup_{i: I} S_i) = \bigcap_{i: I} V_E(S_i)</annotation></semantics></math> by the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a>.</p> </div> <div class="num_prop" id="GaloisClosureOperator"> <h6 id="proposition_11">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/closure+operators">closure operators</a> from <a class="existingWikiWord" href="/nlab/show/Galois+connection">Galois connection</a>)</strong></p> <p>Given a <a class="existingWikiWord" href="/nlab/show/Galois+connection">Galois connection</a> as in def. <a class="maruku-ref" href="#GaloisConnection"></a>, consider the <a class="existingWikiWord" href="/nlab/show/composition">composites</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>E</mi></msub><mo>∘</mo><msub><mi>V</mi> <mi>E</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> I_E \circ V_E \;\colon\; P(X) \longrightarrow P(X) </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>E</mi></msub><mo>∘</mo><msub><mi>I</mi> <mi>E</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> V_E \circ I_E \;\colon\; P(Y) \longrightarrow P(Y) \,. </annotation></semantics></math></div> <p>These satisfy:</p> <ol> <li> <p>For all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>∈</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S \in P(X)</annotation></semantics></math> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>⊂</mo><msub><mi>I</mi> <mi>E</mi></msub><mo>∘</mo><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S \subset I_E \circ V_E(S)</annotation></semantics></math>.</p> </li> <li> <p>For all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>∈</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S \in P(X)</annotation></semantics></math> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>E</mi></msub><mo>∘</mo><msub><mi>I</mi> <mi>E</mi></msub><mo>∘</mo><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V_E \circ I_E \circ V_E (S) = V_E(S)</annotation></semantics></math>.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>E</mi></msub><mo>∘</mo><msub><mi>V</mi> <mi>E</mi></msub></mrow><annotation encoding="application/x-tex">I_E \circ V_E</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/idempotent">idempotent</a> and <a class="existingWikiWord" href="/nlab/show/covariant+functor">covariant</a>.</p> </li> </ol> <p>and</p> <ol> <li> <p>For all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>∈</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T \in P(Y)</annotation></semantics></math> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>⊂</mo><msub><mi>V</mi> <mi>E</mi></msub><mo>∘</mo><msub><mi>I</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T \subset V_E \circ I_E(T)</annotation></semantics></math>.</p> </li> <li> <p>For all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>∈</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T \in P(Y)</annotation></semantics></math> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>E</mi></msub><mo>∘</mo><msub><mi>V</mi> <mi>E</mi></msub><mo>∘</mo><msub><mi>I</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>I</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">I_E \circ V_E \circ I_E (T) = I_E(T)</annotation></semantics></math>.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>E</mi></msub><mo>∘</mo><msub><mi>I</mi> <mi>E</mi></msub></mrow><annotation encoding="application/x-tex">V_E \circ I_E</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/idempotent">idempotent</a> and <a class="existingWikiWord" href="/nlab/show/covariant+functor">covariant</a>.</p> </li> </ol> <p>This is summarized by saying that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>E</mi></msub><mo>∘</mo><msub><mi>V</mi> <mi>E</mi></msub></mrow><annotation encoding="application/x-tex">I_E \circ V_E</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>E</mi></msub><mo>∘</mo><msub><mi>I</mi> <mi>E</mi></msub></mrow><annotation encoding="application/x-tex">V_E \circ I_E</annotation></semantics></math> are <em><a class="existingWikiWord" href="/nlab/show/closure+operators">closure operators</a></em> (<a class="existingWikiWord" href="/nlab/show/idempotent+monads">idempotent monads</a>).</p> </div> <div class="proof"> <h6 id="proof_10">Proof</h6> <p>The first statement is immediate from the adjunction law (prop. <a class="maruku-ref" href="#GaloisConnectionAsAdjunction"></a>).</p> <p>Regarding the second statement: This holds because applied to sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">I_E(T)</annotation></semantics></math>, we see <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>⊂</mo><msub><mi>I</mi> <mi>E</mi></msub><mo>∘</mo><msub><mi>V</mi> <mi>E</mi></msub><mo>∘</mo><msub><mi>I</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">I_E(T) \subset I_E \circ V_E \circ I_E(T)</annotation></semantics></math>. But applying the contravariant map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>E</mi></msub></mrow><annotation encoding="application/x-tex">I_E</annotation></semantics></math> to the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>⊂</mo><msub><mi>V</mi> <mi>E</mi></msub><mo>∘</mo><msub><mi>I</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T \subset V_E \circ I_E(T)</annotation></semantics></math>, we also have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>E</mi></msub><mo>∘</mo><msub><mi>V</mi> <mi>E</mi></msub><mo>∘</mo><msub><mi>I</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>⊂</mo><msub><mi>I</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">I_E \circ V_E \circ I_E(T) \subset I_E(T)</annotation></semantics></math>.</p> <p>This directly implies that the function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>E</mi></msub><mo>∘</mo><msub><mi>V</mi> <mi>E</mi></msub></mrow><annotation encoding="application/x-tex">I_E \circ V_E</annotation></semantics></math>. is idempotent, hence the third statement.</p> <p>The argument for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>E</mi></msub><mo>∘</mo><msub><mi>I</mi> <mi>E</mi></msub></mrow><annotation encoding="application/x-tex">V_E \circ I_E</annotation></semantics></math> is directly analogous.</p> </div> <p>In view of prop. <a class="maruku-ref" href="#GaloisClosureOperator"></a> we say that:</p> <div class="num_defn" id="GaloisClosedElements"> <h6 id="definition_7">Definition</h6> <p><strong>(closed elements)</strong></p> <p>Given a <a class="existingWikiWord" href="/nlab/show/Galois+connection">Galois connection</a> as in def. <a class="maruku-ref" href="#GaloisConnection"></a>, then</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>∈</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S \in P(X)</annotation></semantics></math> is called <em>closed</em> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>E</mi></msub><mo>∘</mo><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>=</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">I_E \circ V_E(S) = S</annotation></semantics></math>;</p> </li> <li> <p>the <em>closure</em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>∈</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S \in P(X)</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cl</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>≔</mo><msub><mi>I</mi> <mi>E</mi></msub><mo>∘</mo><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cl(S) \coloneqq I_E \circ V_E(S)</annotation></semantics></math></p> </li> </ol> <p>and similarly</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>∈</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T \in P(Y)</annotation></semantics></math> is called <em>closed</em> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>E</mi></msub><mo>∘</mo><msub><mi>I</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>=</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">V_E \circ I_E(T) = T</annotation></semantics></math>;</p> </li> <li> <p>the <em>closure</em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>∈</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T \in P(Y)</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cl</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>≔</mo><msub><mi>V</mi> <mi>E</mi></msub><mo>∘</mo><msub><mi>I</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cl(T) \coloneqq V_E \circ I_E(T)</annotation></semantics></math>.</p> </li> </ol> </div> <p>It follows from the properties of <a class="existingWikiWord" href="/nlab/show/closure+operators">closure operators</a>, hence form prop. <a class="maruku-ref" href="#GaloisClosureOperator"></a>:</p> <div class="num_prop" id="GaloisFixedPoints"> <h6 id="proposition_12">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/fixed+point+of+an+adjunction">fixed points</a> of a <a class="existingWikiWord" href="/nlab/show/Galois+connection">Galois connection</a>)</strong></p> <p>Given a <a class="existingWikiWord" href="/nlab/show/Galois+connection">Galois connection</a> as in def. <a class="maruku-ref" href="#GaloisConnection"></a>, then</p> <ol> <li> <p>the closed elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(X)</annotation></semantics></math> are precisely those in the <a class="existingWikiWord" href="/nlab/show/image">image</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>im</mi><mo stretchy="false">(</mo><msub><mi>I</mi> <mi>E</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">im(I_E)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>E</mi></msub></mrow><annotation encoding="application/x-tex">I_E</annotation></semantics></math>;</p> </li> <li> <p>the closed elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(Y)</annotation></semantics></math> are precisely those in the <a class="existingWikiWord" href="/nlab/show/image">image</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>im</mi><mo stretchy="false">(</mo><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">im(V_E)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>E</mi></msub></mrow><annotation encoding="application/x-tex">V_E</annotation></semantics></math>.</p> </li> </ol> <p>We says these are the <em><a class="existingWikiWord" href="/nlab/show/fixed+point+of+an+adjunction">fixed points</a></em> of the Galois connection. Therefore the restriction of the Galois connection</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><munderover><mo>⊥</mo><munder><mo>⟶</mo><mrow><msub><mi>V</mi> <mi>E</mi></msub></mrow></munder><mover><mo>⟵</mo><mrow><msub><mi>I</mi> <mi>E</mi></msub></mrow></mover></munderover><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex"> P(X) \underoverset{\underset{V_E}{\longrightarrow}}{\overset{I_E}{\longleftarrow}}{\bot} P(Y)^{op} </annotation></semantics></math></div> <p>to these fixed points yields an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>im</mi><mo stretchy="false">(</mo><msub><mi>I</mi> <mi>E</mi></msub><mo stretchy="false">)</mo><munderover><mo>≃</mo><munder><mo>⟶</mo><mrow><msub><mi>V</mi> <mi>E</mi></msub></mrow></munder><mover><mo>⟵</mo><mrow><msub><mi>I</mi> <mi>E</mi></msub></mrow></mover></munderover><mi>im</mi><mo stretchy="false">(</mo><msub><mi>V</mi> <mi>E</mi></msub><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex"> im(I_E) \underoverset{\underset{V_E}{\longrightarrow}}{\overset{I_E}{\longleftarrow}}{\simeq} im(V_E)^{op} </annotation></semantics></math></div> <p>now called a <em><a class="existingWikiWord" href="/nlab/show/Galois+correspondence">Galois correspondence</a></em>.</p> </div> <div class="num_prop"> <h6 id="proposition_13">Proposition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/Galois+connection">Galois connection</a> as in def. <a class="maruku-ref" href="#GaloisConnection"></a>, then the sets of closed elements according to def. <a class="maruku-ref" href="#GaloisClosedElements"></a> are closed under forming <a class="existingWikiWord" href="/nlab/show/intersections">intersections</a>.</p> </div> <div class="proof"> <h6 id="proof_11">Proof</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>T</mi> <mi>i</mi></msub><mo>∈</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>:</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{T_i \in P(Y)\}_{i: I}</annotation></semantics></math> is a collection of elements closed under the operator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>=</mo><msub><mi>V</mi> <mi>E</mi></msub><mo>∘</mo><msub><mi>I</mi> <mi>E</mi></msub></mrow><annotation encoding="application/x-tex">K = V_E \circ I_E</annotation></semantics></math>, then by the first item in prop. <a class="maruku-ref" href="#GaloisClosureOperator"></a> it is automatic that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">⋂</mo> <mrow><mi>i</mi><mo>:</mo><mi>I</mi></mrow></msub><msub><mi>T</mi> <mi>i</mi></msub><mo>⊂</mo><mi>K</mi><mo stretchy="false">(</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">⋂</mo> <mrow><mi>i</mi><mo>:</mo><mi>I</mi></mrow></msub><msub><mi>T</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\bigcap_{i: I} T_i \subset K(\bigcap_{i: I} T_i)</annotation></semantics></math>, so it suffices to prove the reverse inclusion. But since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">⋂</mo> <mrow><mi>i</mi><mo>:</mo><mi>I</mi></mrow></msub><msub><mi>T</mi> <mi>i</mi></msub><mo>⊂</mo><msub><mi>T</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\bigcap_{i: I} T_i \subset T_i</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> is covariant and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">T_i</annotation></semantics></math> is closed, we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">⋂</mo> <mrow><mi>i</mi><mo>:</mo><mi>I</mi></mrow></msub><msub><mi>T</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>⊂</mo><mi>K</mi><mo stretchy="false">(</mo><msub><mi>T</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>⊂</mo><msub><mi>T</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">K(\bigcap_{i: I} T_i) \subset K(T_i) \subset T_i</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">⋂</mo> <mrow><mi>i</mi><mo>:</mo><mi>I</mi></mrow></msub><msub><mi>T</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>⊂</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">⋂</mo> <mrow><mi>i</mi><mo>:</mo><mi>I</mi></mrow></msub><msub><mi>T</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">K(\bigcap_{i: I} T_i) \subset \bigcap_{i: I} T_i</annotation></semantics></math> follows.</p> </div> <h3 id="GaloisConnectionAppliedToAffineSpace">Applied to affine space</h3> <p>We now redo the discussion of the Zariski topology on the affine space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">k^n</annotation></semantics></math> from <a href="#OnAffineSpace">above</a> as a special case of the general considerations of <a class="existingWikiWord" href="/nlab/show/Galois+connections">Galois connections</a>.</p> <div class="num_example" id="ZariskiClosedSubsetsInaffineViaGalois"> <h6 id="example_3">Example</h6> <p><strong>(Zariski closed subsets in affine space via Galois connection)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/field">field</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">k[X_1, \cdots, X_n]</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/polynomial+ring">polynomial ring</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/variables">variables</a>. Define a <a class="existingWikiWord" href="/nlab/show/relation">relation</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>↪</mo><mi>k</mi><mo stretchy="false">[</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">]</mo><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex"> E \hookrightarrow k[x_1, \ldots, x_n] \times k^n </annotation></semantics></math></div> <p>by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≔</mo><mrow><mo>(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> E(f, x)\coloneqq \left( f(x) = 0\right) \,. </annotation></semantics></math></div> <p>By def. <a class="maruku-ref" href="#GaloisConnection"></a> and prop. <a class="maruku-ref" href="#GaloisConnectionAsAdjunction"></a> we obtain the corresponding <a class="existingWikiWord" href="/nlab/show/Galois+connection">Galois connection</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">]</mo><mo stretchy="false">)</mo><munderover><mo>⊥</mo><munder><mo>⟶</mo><mrow><msub><mi>V</mi> <mi>E</mi></msub></mrow></munder><mover><mo>⟵</mo><mrow><msub><mi>I</mi> <mi>E</mi></msub></mrow></mover></munderover><mi>P</mi><mo stretchy="false">(</mo><msup><mi>k</mi> <mi>n</mi></msup><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex"> P(k[X_1, \cdots, X_n]) \underoverset{\underset{V_E}{\longrightarrow}}{\overset{I_E}{\longleftarrow}}{\bot} P(k^n)^{op} </annotation></semantics></math></div> <p>(where now <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">k[X_1, \cdots, X_n]</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">k^n</annotation></semantics></math> denote their underlying sets).</p> <p>Here by def. <a class="maruku-ref" href="#GaloisConnection"></a> the function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>E</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>P</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">[</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>⟶</mo><mi>P</mi><mo stretchy="false">(</mo><msup><mi>k</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> V_E \;\colon\; P(k[x_1, \ldots, x_n]) \longrightarrow P(k^n) </annotation></semantics></math></div> <p>sends a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℱ</mi></mrow><annotation encoding="application/x-tex">\mathcal{F}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/polynomials">polynomials</a> to its corresponding <em><a class="existingWikiWord" href="/nlab/show/variety">variety</a></em>,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>∈</mo><msup><mi>k</mi> <mi>n</mi></msup><mspace width="thinmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thinmathspace"></mspace><msub><mo>∀</mo> <mrow><mi>f</mi><mo>∈</mo><mi>k</mi><mo stretchy="false">[</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">]</mo></mrow></msub><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mi>f</mi><mo>∈</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mo>⇒</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> V_E(\mathcal{F}) = \{\vec x \in k^n \,\vert\, \forall_{f \in k[x_1, \ldots, x_n]} \; (f \in \mathcal{F}) \Rightarrow (f(x) = 0)\} \,. </annotation></semantics></math></div> <p>These are just the Zariski closed subsets from def. <a class="maruku-ref" href="#ZariskiOpenSubsetsOnAffineSpace"></a>.</p> <p>In the other direction,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>E</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>P</mi><mo stretchy="false">(</mo><msup><mi>k</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>⟶</mo><mi>P</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">[</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> I_E \;\colon\; P(k^n) \longrightarrow P(k[x_1, \ldots, x_n]) </annotation></semantics></math></div> <p>sends a set of points <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>⊆</mo><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">T \subseteq k^n</annotation></semantics></math> to its corresponding <em><a class="existingWikiWord" href="/nlab/show/vanishing+ideal">vanishing ideal</a></em></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mi>f</mi><mo>∈</mo><mi>k</mi><mo stretchy="false">[</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thinmathspace"></mspace><msub><mo>∀</mo> <mrow><mi>x</mi><mo>:</mo><msup><mi>k</mi> <mi>n</mi></msup></mrow></msub><mspace width="thickmathspace"></mspace><mi>x</mi><mo>∈</mo><mi>T</mi><mo>⇒</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> I_E(T) = \{f \in k[x_1, \ldots, x_n] \,\vert\, \forall_{x: k^n} \; x \in T \Rightarrow f(x) = 0\} </annotation></semantics></math></div> <p>which we considered earlier in def. <a class="maruku-ref" href="#VanishingIdeal"></a>.</p> </div> <p>We may now use the abstract theory of Galois connections to verify that Zariski closed subsets form a <a class="existingWikiWord" href="/nlab/show/topological+space">topology</a>:</p> <div class="num_prop" id="ZariskiTopologyOnAffineSpaceViaGaloisConnectionWellDefined"> <h6 id="proposition_14">Proposition</h6> <p><strong>(Zariski topology is well defined)</strong></p> <p>Using <a class="existingWikiWord" href="/nlab/show/excluded+middle">excluded middle</a>, then:</p> <p>The set of Zariski closed subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">k^n</annotation></semantics></math> from example <a class="maruku-ref" href="#ZariskiClosedSubsetsInaffineViaGalois"></a> constitutes a <a class="existingWikiWord" href="/nlab/show/topological+space">topology</a> in that it is closed under</p> <ol> <li> <p>arbitrary intersections;</p> </li> <li> <p>finite untions.</p> </li> </ol> </div> <div class="proof"> <h6 id="proof_12">Proof</h6> <p>Regarding the first point: From prop. <a class="maruku-ref" href="#GaloisConnectionAsAdjunction"></a> we know that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>E</mi></msub></mrow><annotation encoding="application/x-tex">V_E</annotation></semantics></math> takes unions to intersections, hence that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo>∩</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><msub><mi>ℱ</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msub><mi>V</mi> <mi>E</mi></msub><mrow><mo>(</mo><munder><mo>∪</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>ℱ</mi> <mi>i</mi></msub><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \underset{i \in I}{\cap} V_E(\mathcal{F}_i) \;=\; V_E\left( \underset{i \in I}{\cup} \mathcal{F}_i \right) \,. </annotation></semantics></math></div> <p>Regarding the second point, we exploit the <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> structure of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">[</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">k[x_1, \ldots, x_n]</annotation></semantics></math>. It is sufficient to show that the set of Zariski closed sets is closed under the empty union and under binary unions.</p> <p>The empty union is the entire space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">k^n</annotation></semantics></math>, which is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V(1)</annotation></semantics></math> (the variety associated with the constant polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>),</p> <p>Hence it only remains to see closure under binary unions.</p> <p>To this end, recall from prop. <a class="maruku-ref" href="#GaloisClosureOperator"></a> that we may replace <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℱ</mi></mrow><annotation encoding="application/x-tex">\mathcal{F}</annotation></semantics></math> with the corresponding ideal</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>≔</mo><msub><mi>I</mi> <mi>E</mi></msub><mo>∘</mo><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> I \coloneqq I_E \circ V_E(\mathcal{F}) </annotation></semantics></math></div> <p>without changing the variety:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>ℱ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> V_E(I) = V_E(\mathcal{F}) \,. </annotation></semantics></math></div> <p>With this it is sufficient to show that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo>∪</mo><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>I</mi><mo>′</mo><mo stretchy="false">)</mo><mo>=</mo><mi>V</mi><mo stretchy="false">(</mo><mi>I</mi><mo>⋅</mo><mi>I</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> V_E(I) \cup V_E(I') = V(I \cdot I') </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>⋅</mo><mi>I</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">I \cdot I'</annotation></semantics></math> is the ideal consisting of finite sums of elements of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mi>g</mi></mrow><annotation encoding="application/x-tex">f g</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">f \in I</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∈</mo><mi>I</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">g \in I'</annotation></semantics></math>.</p> <p>We conclude by proving this statement:</p> <p>Applying the contravariant operator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>E</mi></msub></mrow><annotation encoding="application/x-tex">V_E</annotation></semantics></math> to the inclusions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>⋅</mo><mi>I</mi><mo>′</mo><mo>⊆</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">I \cdot I' \subseteq I</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>⋅</mo><mi>I</mi><mo>′</mo><mi>subseteq</mi><mi>I</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">I \cdot I' subseteq I'</annotation></semantics></math> (which are clear since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>,</mo><mi>I</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">I, I'</annotation></semantics></math> are ideals), we derive <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo>⊆</mo><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>I</mi><mo>⋅</mo><mi>I</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V_E(I) \subseteq V_E(I \cdot I')</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>I</mi><mo>′</mo><mo stretchy="false">)</mo><mo>⊆</mo><mi>V</mi><mo stretchy="false">(</mo><mi>I</mi><mo>⋅</mo><mi>I</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V_E(I') \subseteq V(I \cdot I')</annotation></semantics></math>, so the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo>∪</mo><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>I</mi><mo>′</mo><mo stretchy="false">)</mo><mo>⊆</mo><mi>V</mi><mo stretchy="false">(</mo><mi>I</mi><mo>⋅</mo><mi>I</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V_E(I) \cup V_E(I') \subseteq V(I \cdot I')</annotation></semantics></math> is automatic.</p> <p>In the other direction, to prove <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>I</mi><mo>⋅</mo><mi>I</mi><mo>′</mo><mo stretchy="false">)</mo><mo>⊆</mo><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo>∪</mo><mi>V</mi><mo stretchy="false">(</mo><mi>I</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V(I \cdot I') \subseteq V_E(I) \cup V(I')</annotation></semantics></math>, suppose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>V</mi><mo stretchy="false">(</mo><mi>I</mi><mo>⋅</mo><mi>I</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x \in V(I \cdot I')</annotation></semantics></math> and that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> <em>doesn’t</em> belong to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V(I)</annotation></semantics></math>. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f(x) \neq 0</annotation></semantics></math> for some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">f \in I</annotation></semantics></math>. For every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∈</mo><mi>I</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">g \in I'</annotation></semantics></math>, we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>f</mi><mo>⋅</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f(x)g(x) = (f \cdot g)(x) = 0</annotation></semantics></math> since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>⋅</mo><mi>g</mi><mo>∈</mo><mi>I</mi><mo>⋅</mo><mi>I</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">f \cdot g \in I \cdot I'</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>I</mi><mo>⋅</mo><mi>I</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x \in V_E(I \cdot I')</annotation></semantics></math>. Now divide by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math> to get <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">g(x) = 0</annotation></semantics></math> for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∈</mo><mi>I</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">g \in I'</annotation></semantics></math>, so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>I</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x \in V_E(I')</annotation></semantics></math>.</p> </div> <div class="num_example" id="ZariskiTopologyOnMaximalIdealsOfPolynomialRingViaGaloisConnection"> <h6 id="example_4">Example</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/field">field</a>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">k[X_1, \cdots, X_n]</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/polynomial+ring">polynomial ring</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/variables">variables</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MaxIdl</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">MaxIdl(k[X_1, \cdots, X_n])</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/set">set</a> of <a class="existingWikiWord" href="/nlab/show/maximal+ideals">maximal ideals</a> in this ring.</p> <p>Define then a <a class="existingWikiWord" href="/nlab/show/relation">relation</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>↪</mo><mi>k</mi><mo stretchy="false">[</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">]</mo><mo>×</mo><mi>MaxIdeal</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">[</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> E \hookrightarrow k[x_1, \ldots, x_n] \times MaxIdeal(k[x_1, \ldots, x_n]) </annotation></semantics></math></div> <p>by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>M</mi><mo stretchy="false">)</mo><mo>⇔</mo><mo stretchy="false">(</mo><mi>f</mi><mo>∈</mo><mi>M</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> E(f, M) \Leftrightarrow (f \in M) \,. </annotation></semantics></math></div> <p>For a subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>⊆</mo><mi>MaxIdl</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">[</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T \subseteq MaxIdl(k[x_1, \ldots, x_n])</annotation></semantics></math> we calculate</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mi>f</mi><mo>∈</mo><mi>k</mi><mo stretchy="false">[</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">]</mo><mo>:</mo><msub><mo>∀</mo> <mrow><mi>𝔪</mi><mo>∈</mo><mi>MaxIdl</mi></mrow></msub><mi>M</mi><mo>∈</mo><mi>S</mi><mo>⇒</mo><mi>f</mi><mo>∈</mo><mi>𝔪</mi><mo stretchy="false">}</mo><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">⋂</mo> <mrow><mi>𝔪</mi><mo>∈</mo><mi>S</mi></mrow></munder><mi>𝔪</mi></mrow><annotation encoding="application/x-tex"> I_E(T) = \{f \in k[x_1, \ldots, x_n]: \forall_{\mathfrak{m} \in MaxIdl} M \in S \Rightarrow f \in \mathfrak{m}\} = \bigcap_{\mathfrak{m} \in S} \mathfrak{m} </annotation></semantics></math></div> <p>which is an ideal, since the intersection of any collection of ideals is again an ideal. (However, not all ideals are given as intersections of maximal ideals, a point to which we will return in a moment.)</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>This is a slight generalization of example <a class="maruku-ref" href="#ZariskiClosedSubsetsInaffineViaGalois"></a> since each point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a = (a_1, \ldots, a_n)</annotation></semantics></math> induces a maximal ideal</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝔪</mi> <mi>a</mi></msub><mo>≔</mo><mo stretchy="false">⟨</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>−</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo>−</mo><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy="false">⟩</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathfrak{m}_a \coloneqq \langle x_1 - a_1, \ldots, x_n - a_n \rangle \,, </annotation></semantics></math></div> <p>i.e. the <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> of the function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>k</mi><mo stretchy="false">[</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">]</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>k</mi></mtd></mtr> <mtr><mtd><mi>f</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ k[x_1, \ldots, x_n] &\longrightarrow& k \\ f &\mapsto& f(a) } </annotation></semantics></math></div> <p>which evaluates polynomials <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> at the point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math>, where we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f(a) = 0</annotation></semantics></math> iff <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><msub><mi>𝔪</mi> <mi>a</mi></msub></mrow><annotation encoding="application/x-tex">f \in \mathfrak{m}_a</annotation></semantics></math>.</p> <p>Of course it need not be the case that all maximal ideals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔪</mi></mrow><annotation encoding="application/x-tex">\mathfrak{m}</annotation></semantics></math> are given by points in this way; for example, the ideal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>2</mn></msup><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x^2 + 1)</annotation></semantics></math> is maximal in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mo stretchy="false">[</mo><mi>x</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}[x]</annotation></semantics></math> but is not given by evaluation at a point because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">x^2 + 1</annotation></semantics></math> does not vanish at any real point. However, if the <a class="existingWikiWord" href="/nlab/show/ground+field">ground field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/algebraically+closed+field">algebraically closed</a>, then every maximal ideal of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">[</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">k[x_1, \ldots, x_n]</annotation></semantics></math> is given by evaluation at a point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a = (a_1, \ldots, a_n)</annotation></semantics></math>. This result is not completely obvious; it is sometimes called the <em>weak <a class="existingWikiWord" href="/nlab/show/Nullstellensatz">Nullstellensatz</a></em>.</p> </div> <div class="num_prop"> <h6 id="proposition_15">Proposition</h6> <p>The set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>⊆</mo><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">S \subseteq k^n</annotation></semantics></math> that are closed under the operator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>E</mi></msub><mo>∘</mo><msub><mi>I</mi> <mi>E</mi></msub><mo>:</mo><mi>P</mi><mo stretchy="false">(</mo><msup><mi>k</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>→</mo><mi>P</mi><mo stretchy="false">(</mo><msup><mi>k</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V_E \circ I_E: P(k^n) \to P(k^n)</annotation></semantics></math> in example <a class="maruku-ref" href="#ZariskiTopologyOnMaximalIdealsOfPolynomialRingViaGaloisConnection"></a> form a <a class="existingWikiWord" href="/nlab/show/topology">topology</a>.</p> </div> <div class="proof"> <h6 id="proof_13">Proof</h6> <p>The proof is virtually the same as in the proof of prop. <a class="maruku-ref" href="#ZariskiTopologyOnAffineSpaceViaGaloisConnectionWellDefined"></a>: they are closed under arbitrary intersections by our earlier generalities, and they are closed under finite unions by the similar reasoning: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V_E(S) = V_E(I)</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>=</mo><msub><mi>I</mi> <mi>E</mi></msub><mo>∘</mo><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">I = I_E \circ V_E(S)</annotation></semantics></math> is an ideal, so there is no loss of generality in considering <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V_E(I)</annotation></semantics></math> for ideals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo>∪</mo><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>I</mi><mo>′</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>I</mi><mo>⋅</mo><mi>I</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V_E(I) \cup V_E(I') = V_E(I \cdot I')</annotation></semantics></math>. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔪</mi><mo>∈</mo><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>I</mi><mo>⋅</mo><mi>I</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{m} \in V_E(I \cdot I')</annotation></semantics></math> (meaning <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>⋅</mo><mi>I</mi><mo>′</mo><mo>⊆</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">I \cdot I' \subseteq M</annotation></semantics></math>) but <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔪</mi></mrow><annotation encoding="application/x-tex">\mathfrak{m}</annotation></semantics></math> <em>doesn’t</em> belong to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V_E(I)</annotation></semantics></math>, i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∉</mo><mi>𝔪</mi></mrow><annotation encoding="application/x-tex">f \notin \mathfrak{m}</annotation></semantics></math> for some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">f \in I</annotation></semantics></math>, then for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∈</mo><mi>I</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">g \in I'</annotation></semantics></math> we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>⋅</mo><mi>g</mi><mo>∈</mo><mi>𝔪</mi></mrow><annotation encoding="application/x-tex">f \cdot g \in \mathfrak{m}</annotation></semantics></math>. Taking the <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo>:</mo><mi>R</mi><mo>→</mo><mi>R</mi><mo stretchy="false">/</mo><mi>𝔪</mi></mrow><annotation encoding="application/x-tex">\pi: R \to R/\mathfrak{m}</annotation></semantics></math> to the field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">/</mo><mi>𝔪</mi></mrow><annotation encoding="application/x-tex">R/\mathfrak{m}</annotation></semantics></math>, we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo stretchy="false">(</mo><mi>f</mi><mo>⋅</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mi>π</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>π</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\pi(f \cdot g) = \pi(f)\cdot \pi(g) = 0</annotation></semantics></math>, and since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\pi(f) \neq 0</annotation></semantics></math> we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\pi(g) = 0</annotation></semantics></math> for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∈</mo><mi>I</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">g \in I'</annotation></semantics></math>, hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔪</mi><mo>∈</mo><msub><mi>V</mi> <mi>E</mi></msub><mo stretchy="false">(</mo><mi>I</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{m} \in V_E(I')</annotation></semantics></math>.</p> </div> <p>Thus the fixed elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>E</mi></msub><mo>∘</mo><msub><mi>I</mi> <mi>E</mi></msub></mrow><annotation encoding="application/x-tex">V_E \circ I_E</annotation></semantics></math> on one side of the Galois correspondence are the closed sets of a topology. The fixed elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>E</mi></msub><mo>∘</mo><msub><mi>V</mi> <mi>E</mi></msub></mrow><annotation encoding="application/x-tex">I_E \circ V_E</annotation></semantics></math> on the other side are a matter of interest; in the case where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/algebraically+closed+field">algebraically closed</a>, they are the <em><a class="existingWikiWord" href="/nlab/show/radical+ideals">radical ideals</a></em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">k[X_1, \ldots, X_n]</annotation></semantics></math> according to the “strong” <a class="existingWikiWord" href="/nlab/show/Nullstellensatz">Nullstellensatz</a>.</p> <h3 id="GaloisAppliedToAffineSchemes">Applied to affine schemes</h3> <p>We now redo the discussion of the Zariski topology on the <a class="existingWikiWord" href="/nlab/show/prime+spectrum+of+a+commutative+ring">prime spectrum of a commutative ring</a> from <a href="#OnAffineVarieties">above</a> as a special case of the general considerations of <a class="existingWikiWord" href="/nlab/show/Galois+connections">Galois connections</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Zariski+site">Zariski site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/schemes+are+sober">schemes are sober</a></p> </li> </ul> <h2 id="references">References</h2> <p>Lecture notes include</p> <ul> <li>Jim Carrell, <em>Zariski topology</em> <a href="https://personal.math.ubc.ca/~carrell/423.pdf">pdf</a></li> </ul> <p>See also</p> <ul> <li>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Zariski_topology">Zariski topology</a></em></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on December 19, 2024 at 01:10:47. 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