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It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" 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"> <p class="show_diff"> Showing changes from revision #17 to #18: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <ins class='diffins'><div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='compact_objects'>Compact objects</h4> <div class='hide'> <p><strong>objects <math class='maruku-mathml' display='inline' id='mathml_26576585ec35bbdafa9da596de5e82f522370ed3_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>d \in C</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_26576585ec35bbdafa9da596de5e82f522370ed3_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo stretchy='false'>(</mo><mi>d</mi><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>C(d,-)</annotation></semantics></math> commutes with certain <a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimits</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/coproduct-preserving+representable'>coproduct-preserving representable</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/connected+object'>connected object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+object'>compact object</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+element'>compact object in a (0,1)-category</a>, <a class='existingWikiWord' href='/nlab/show/diff/compact+object+in+an+%28infinity%2C1%29-category'>compact object in an (∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/finite+object'>finite object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/small+object'>small object</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/tiny+object'>tiny object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/infinitesimal+object'>atomic object</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/amazing+right+adjoint'>amazing right adjoint</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/accessible+category'>accessible category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+category'>presentable category</a></li> </ul> </li> </ul> <h2 id='models'>Models</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/proper+geometric+morphism'>compact topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+space'>compact topological space</a></p> </li> </ul> <h2 id='relative_version'>Relative version</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/proper+geometric+morphism'>proper geometric morphism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/proper+map'>proper map</a></p> </li> </ul> <div> <p> <a href='/nlab/edit/compact+object+-+contents'>Edit this sidebar</a> </p> </div></div> </div> </div></ins><ins class='diffins'> </ins><h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#definition'>Definition</a></li><li><a href='#properties'>Properties</a></li><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 modifier <strong>‘quasi-compact’</strong> (or simply <strong>‘quasicompact’</strong>) is used to denote a compactness property in the relative setup (that is, for morphisms) and in the setups emphasising non-Hausdorff topology. For example, schemes over complex numbers have both the <a class='existingWikiWord' href='/nlab/show/diff/complex+analytic+topology'>complex analytic topology</a> and the <a class='existingWikiWord' href='/nlab/show/diff/Zariski+topology'>Zariski topology</a>; we use “quasicompact” for Zariski and “compact” for complex topology in that setup.</p> <p>For many topologists, analysts and so on, the word ‘compact’ means a <a class='existingWikiWord' href='/nlab/show/diff/compactum'>compact Hausdorff space</a>. Some topological schools distinguish a “<a class='existingWikiWord' href='/nlab/show/diff/compact+space'>compact space</a>” (which is not necessarily <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff</a>) and a “<a class='existingWikiWord' href='/nlab/show/diff/compactum'>compactum</a>” (which is Hausdorff); and similarly a “<a class='existingWikiWord' href='/nlab/show/diff/paracompact+topological+space'>paracompact space</a>” and a “<a class='existingWikiWord' href='/nlab/show/diff/paracompact+topological+space'>paracompactum</a>”. In <a class='existingWikiWord' href='/nlab/show/diff/algebraic+geometry'>algebraic geometry</a>, in contrast, one usually says <strong>quasicompact space</strong> to denote a topological space which is compact but not necessarily Hausdorff. For example, the <a class='existingWikiWord' href='/nlab/show/diff/Zariski+topology'>Zariski topology</a> on an <a class='existingWikiWord' href='/nlab/show/diff/algebraic+variety'>algebraic variety</a> and the topology of an étalé space of a <a class='existingWikiWord' href='/nlab/show/diff/sheaf'>sheaf</a> (even over a Hausdorff topological space) are typically not Hausdorff.</p> <h2 id='definition'>Definition</h2> <p>A <a class='existingWikiWord' href='/nlab/show/diff/scheme'>scheme</a> is quasicompact iff it has a Zariski cover by <a class='existingWikiWord' href='/nlab/show/diff/finite+number'>finitely many</a> open affine subschemes. In particular, any affine scheme is quasicompact.</p> <p>Most important is the relative version of this concept. A morphism <math class='maruku-mathml' display='inline' id='mathml_a6b2da3468ee20fdc9c9d96af253270a72deb8e3_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo lspace='verythinmathspace'>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>f\colon X \to Y</annotation></semantics></math> of schemes is a <strong>quasicompact morphism</strong> if the inverse image of a quasicompact Zariski open subset of <math class='maruku-mathml' display='inline' id='mathml_a6b2da3468ee20fdc9c9d96af253270a72deb8e3_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> is quasicompact (EGAI6.6.1).</p> <p>It is straightforward to show [EGAI6.6.4] that it is enough to require this for affine subsets of <math class='maruku-mathml' display='inline' id='mathml_a6b2da3468ee20fdc9c9d96af253270a72deb8e3_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math>, or even to require the existence of a single covering <math class='maruku-mathml' display='inline' id='mathml_a6b2da3468ee20fdc9c9d96af253270a72deb8e3_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi><mo>=</mo><msub><mo>∪</mo> <mi>i</mi></msub><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>Y = \cup_i U_i</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_a6b2da3468ee20fdc9c9d96af253270a72deb8e3_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> by open affine subschemes <math class='maruku-mathml' display='inline' id='mathml_a6b2da3468ee20fdc9c9d96af253270a72deb8e3_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>U_i</annotation></semantics></math>, such that the inverse image <math class='maruku-mathml' display='inline' id='mathml_a6b2da3468ee20fdc9c9d96af253270a72deb8e3_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><msub><mo>×</mo> <mi>Y</mi></msub><mi>X</mi></mrow><annotation encoding='application/x-tex'>U_i \times_Y X</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_a6b2da3468ee20fdc9c9d96af253270a72deb8e3_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>U_i</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_a6b2da3468ee20fdc9c9d96af253270a72deb8e3_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is quasicompact. A scheme <math class='maruku-mathml' display='inline' id='mathml_a6b2da3468ee20fdc9c9d96af253270a72deb8e3_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> over a base scheme <math class='maruku-mathml' display='inline' id='mathml_a6b2da3468ee20fdc9c9d96af253270a72deb8e3_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> is quasicompact if the canonical morphism <math class='maruku-mathml' display='inline' id='mathml_a6b2da3468ee20fdc9c9d96af253270a72deb8e3_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding='application/x-tex'>X \to S</annotation></semantics></math> is quasicompact. This is consistent, because if <math class='maruku-mathml' display='inline' id='mathml_a6b2da3468ee20fdc9c9d96af253270a72deb8e3_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is a usual scheme (over the spectrum of integers <math class='maruku-mathml' display='inline' id='mathml_a6b2da3468ee20fdc9c9d96af253270a72deb8e3_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo>=</mo><mi>ℤ</mi></mrow><annotation encoding='application/x-tex'>S = \mathbb{Z}</annotation></semantics></math>) or, more generally, a relative scheme over an <em>affine</em> scheme <math class='maruku-mathml' display='inline' id='mathml_a6b2da3468ee20fdc9c9d96af253270a72deb8e3_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>, quasicompactness of the canonical morphism <math class='maruku-mathml' display='inline' id='mathml_a6b2da3468ee20fdc9c9d96af253270a72deb8e3_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding='application/x-tex'>X\to S</annotation></semantics></math> is by the above criteria clearly equivalent to the usual quasicompactness of <math class='maruku-mathml' display='inline' id='mathml_a6b2da3468ee20fdc9c9d96af253270a72deb8e3_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>.</p> <h2 id='properties'>Properties</h2> <p>A composition of quasicompact morphisms is quasicompact, and the pullback of quasicompact morphisms is quasicompact. This enables the definition for <a class='existingWikiWord' href='/nlab/show/diff/algebraic+stack'>algebraic stack</a>s: a morphism of algebraic stacks is quasicompact if the pullback of that morphism to some atlas is quasicompact.</p> <p>Algebraic geometers sometimes (but more rarely) also talk about <strong>quasicompact objects</strong> in more general categories, meaning <a class='existingWikiWord' href='/nlab/show/diff/compact+object'>compact object</a>s (object which corepresent covariant functors commuting with filtered colimits); with or without a modifier denoting a cardinal (<math class='maruku-mathml' display='inline' id='mathml_a6b2da3468ee20fdc9c9d96af253270a72deb8e3_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>κ</mi></mrow><annotation encoding='application/x-tex'>\kappa</annotation></semantics></math>-quasicompact objects).</p> <h2 id='related_concepts'>Related concepts</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/morphism+of+finite+presentation'>finitely presented morphism</a></li> </ul> <h2 id='references'>References</h2> <p>An old discussion on the terminological aspects (Mike, Zoran, Toby) is at <math class='maruku-mathml' display='inline' id='mathml_a6b2da3468ee20fdc9c9d96af253270a72deb8e3_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>Forum <a href='http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=2680&Focus=22689#Comment_22689'>here</a>.</p> <p><div class='property'> category: <a class='category_link' href='/nlab/list/algebraic+geometry'>algebraic geometry</a></div> </p> <p> </p> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on February 3, 2014 at 15:19:07. 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