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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> <h4 id="homological_algebra">Homological algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a></strong></p> <p>(also <a class="existingWikiWord" href="/nlab/show/nonabelian+homological+algebra">nonabelian homological algebra</a>)</p> <p><em><a class="existingWikiWord" href="/schreiber/show/Introduction+to+Homological+Algebra">Introduction</a></em></p> <p><strong>Context</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+and+abelian+categories">additive and abelian categories</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ab-enriched+category">Ab-enriched category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-additive+category">pre-additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-abelian+category">pre-abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+category">Grothendieck category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaves">abelian sheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semi-abelian+category">semi-abelian category</a></p> </li> </ul> <p><strong>Basic definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/kernel">kernel</a>, <a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex">complex</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential">differential</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homology">homology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+map">chain map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+homotopy">chain homotopy</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+homology+and+cohomology">chain homology and cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+resolution">homological resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+on+a+simplicial+set">simplicial homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+homology">generalized homology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exact+sequence">exact sequence</a>,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a>, <a class="existingWikiWord" href="/nlab/show/long+exact+sequence">long exact sequence</a>, <a class="existingWikiWord" href="/nlab/show/split+exact+sequence">split exact sequence</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+object">injective object</a>, <a class="existingWikiWord" href="/nlab/show/projective+object">projective object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+resolution">injective resolution</a>, <a class="existingWikiWord" href="/nlab/show/projective+resolution">projective resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/flat+resolution">flat resolution</a></p> </li> </ul> </li> </ul> <p><strong>Stable homotopy theory notions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a>, <a class="existingWikiWord" href="/nlab/show/enhanced+triangulated+category">enhanced triangulated category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+model+category">stable model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pretriangulated+dg-category">pretriangulated dg-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E-category">A-∞-category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+chain+complexes">(∞,1)-category of chain complexes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a>, <a class="existingWikiWord" href="/nlab/show/derived+functor+in+homological+algebra">derived functor in homological algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Tor">Tor</a>, <a class="existingWikiWord" href="/nlab/show/Ext">Ext</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lim%5E1+and+Milnor+sequences">lim^1 and Milnor sequences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fr-code">fr-code</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> </li> </ul> <p><strong>Constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/double+complex">double complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Koszul-Tate+resolution">Koszul-Tate resolution</a>, <a class="existingWikiWord" href="/nlab/show/BRST-BV+complex">BRST-BV complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+filtered+complex">spectral sequence of a filtered complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+double+complex">spectral sequence of a double complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+spectral+sequence">Grothendieck spectral sequence</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Leray+spectral+sequence">Leray spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Serre+spectral+sequence">Serre spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild-Serre+spectral+sequence">Hochschild-Serre spectral sequence</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong>Lemmas</strong></p> <p><a class="existingWikiWord" href="/nlab/show/diagram+chasing">diagram chasing</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/3x3+lemma">3x3 lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/four+lemma">four lemma</a>, <a class="existingWikiWord" href="/nlab/show/five+lemma">five lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/snake+lemma">snake lemma</a>, <a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/horseshoe+lemma">horseshoe lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Baer%27s+criterion">Baer's criterion</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/Schanuel%27s+lemma">Schanuel's lemma</a></p> <p><strong>Homology theories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cyclic+homology">cyclic homology</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> / <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal</a>, <a class="existingWikiWord" href="/nlab/show/operadic+Dold-Kan+correspondence">operadic</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Moore+complex">Moore complex</a>, <a class="existingWikiWord" href="/nlab/show/Alexander-Whitney+map">Alexander-Whitney map</a>, <a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+map">Eilenberg-Zilber map</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+theorem">Eilenberg-Zilber theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cochain+on+a+simplicial+set">cochain on a simplicial set</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> </ul> </div></div> </div> </div> <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='#examples'>Examples</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 <em>Koszul complex</em> of a sequence of elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>x</mi> <mi>d</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x_1, \cdots, x_d)</annotation></semantics></math> in a <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> (or more generally of <a class="existingWikiWord" href="/nlab/show/center">central</a> elements in a <a class="existingWikiWord" href="/nlab/show/non-commutative+ring">non-commutative ring</a>) is a <a class="existingWikiWord" href="/nlab/show/cochain+complex">cochain complex</a> whose entry in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">-n</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/exterior+power">exterior power</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∧</mo> <mi>n</mi></msup><msup><mi>R</mi> <mi>d</mi></msup></mrow><annotation encoding="application/x-tex">\wedge^n R^d</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/free+module">free module</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>R</mi> <mi>d</mi></msup><mo>=</mo><msup><mi>R</mi> <mrow><msub><mo>⊕</mo> <mi>d</mi></msub></mrow></msup></mrow><annotation encoding="application/x-tex">R^d = R^{\oplus_d}</annotation></semantics></math> over <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> of <a class="existingWikiWord" href="/nlab/show/rank">rank</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>, and whose <a class="existingWikiWord" href="/nlab/show/differential">differential</a> is given in each degree on the <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>th summand by multiplication with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">x_k</annotation></semantics></math>.</p> <p>The key property of the Koszul complex is that in good cases (namely if the sequence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>x</mi> <mi>d</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x_1, \cdots, x_d)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/regular+sequence">regular sequence</a> 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>), it constitutes is a <a class="existingWikiWord" href="/nlab/show/free+resolution">free</a> <a class="existingWikiWord" href="/nlab/show/homological+resolution">homological resolution</a> of the <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>x</mi> <mi>d</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R/(x_1, \cdots, x_d)</annotation></semantics></math> 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> by the <a class="existingWikiWord" href="/nlab/show/ideal">ideal</a> generated by these elements (see prop. <a class="maruku-ref" href="#KoszulComplexOfRegularSequenceIsFreeResolutionOfQuotientRing"></a> below).</p> <p>In cases where the Koszul complex fails to be a <a class="existingWikiWord" href="/nlab/show/homological+resolution">homological resolution</a> of the <a class="existingWikiWord" href="/nlab/show/quotient+ring">quotient ring</a>, it may be augmented by further generators to yield a resolution after all then called a <em><a class="existingWikiWord" href="/nlab/show/Koszul-Tate+resolution">Koszul-Tate resolution</a></em>.</p> <p>From the perspective of <a class="existingWikiWord" href="/nlab/show/derived+algebraic+geometry">derived algebraic geometry</a> the Koszul complex may be interpreted as the <a class="existingWikiWord" href="/nlab/show/formal+duality">formal dual</a> of the <a class="existingWikiWord" href="/nlab/show/derived+critical+locus">derived critical locus</a> of the elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>x</mi> <mi>d</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x_1, \cdots, x_d)</annotation></semantics></math>, regarded as functions on the <a class="existingWikiWord" href="/nlab/show/spectrum+of+a+ring">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>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(R)</annotation></semantics></math>.</p> <p>In this guise the Koszul complex appears prominently in <a class="existingWikiWord" href="/nlab/show/Lagrangian+field+theory">Lagrangian field theory</a>, under the name <em><a class="existingWikiWord" href="/nlab/show/BV-complex">BV-complex</a></em>, as a potential <a class="existingWikiWord" href="/nlab/show/homological+resolution">homological resolution</a> of the <em><a class="existingWikiWord" href="/nlab/show/shell">shell</a></em> (the solution locus of the <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange</a> <a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a>). In this case the obstruction to the Koszul complex providing a resolution of the shell is its cochain cohomology in degree -1 (via prop. <a class="maruku-ref" href="#KoszulResolutionForNoetherianRngAndElementsInJacobson"></a> below) which has the interpretation as the <a class="existingWikiWord" href="/nlab/show/infinitesimal+gauge+symmetries">infinitesimal gauge symmetries</a> of the <a class="existingWikiWord" href="/nlab/show/Lagrangian+density">Lagrangian density</a> that have not been made explicit. Making them explicit by promoting them to elements in the <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a> of the corresponding <a class="existingWikiWord" href="/nlab/show/action+Lie+algebroid">action Lie algebroid</a> yields what is called the <em><a class="existingWikiWord" href="/nlab/show/BRST+complex">BRST complex</a></em> of the theory, and <em>its</em> Koszul complex then yields the respective <a class="existingWikiWord" href="/nlab/show/Koszul-Tate+resolution">Koszul-Tate resolution</a>, now called the <em><a class="existingWikiWord" href="/nlab/show/BV-BRST+complex">BV-BRST complex</a></em> of the theory.</p> <h2 id="definition">Definition</h2> <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/unital+ring">unital ring</a>.</p> <p>Consider also a finite <a class="existingWikiWord" href="/nlab/show/sequence">sequence</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>r</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x_1,\ldots,x_r)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/elements">elements</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mi>i</mi></msub><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">x_i \in R</annotation></semantics></math>.</p> <p>Given any <a class="existingWikiWord" href="/nlab/show/center">central element</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>Z</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x\in Z(R)</annotation></semantics></math>, one can define a two term <a class="existingWikiWord" href="/nlab/show/cochain+complex">cochain complex</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">(</mo><mn>0</mn><mo>→</mo><mi>R</mi><mover><mo>→</mo><mi>x</mi></mover><mi>R</mi><mo>→</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> K(x) \coloneqq (0\to R\stackrel{x}\to R\to 0) </annotation></semantics></math></div> <p>concentrated in degrees <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math> and <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>, where the map (the <a class="existingWikiWord" href="/nlab/show/differential">differential</a>) is the left multiplication by <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>. Given a sequence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>r</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x_1,\ldots,x_r)</annotation></semantics></math> of central elements 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> one can define the tensor product</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" 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>r</mi></msub><mo stretchy="false">)</mo><mo>≔</mo><mi>K</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>R</mi></msub><mi>K</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>R</mi></msub><mi>⋯</mi><msub><mo>⊗</mo> <mi>R</mi></msub><mi>K</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mi>r</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> K(x_1,\ldots,x_r) \coloneqq K(x_1)\otimes_R K(x_2)\otimes_R\cdots \otimes_R K(x_r) </annotation></semantics></math></div> <p>of complexes of left <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 class="existingWikiWord" href="/nlab/show/modules">modules</a>. The degree <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> part 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>r</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(x_1,\ldots,x_r)</annotation></semantics></math> equals the <a class="existingWikiWord" href="/nlab/show/exterior+power">exterior power</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><msup><mi>R</mi> <mi>r</mi></msup></mrow><annotation encoding="application/x-tex">\Lambda^{p+1}R^r</annotation></semantics></math>. Consider the usual <a class="existingWikiWord" href="/nlab/show/basis">basis</a> elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></msub><mo>∧</mo><mi>⋯</mi><mo>∧</mo><msub><mi>e</mi> <mrow><msub><mi>i</mi> <mi>p</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">e_{i_0}\wedge \cdots \wedge e_{i_p}</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><msup><mi>R</mi> <mi>r</mi></msup></mrow><annotation encoding="application/x-tex">\Lambda^{p+1}R^r</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>≤</mo><msub><mi>i</mi> <mn>0</mn></msub><mo><</mo><msub><mi>i</mi> <mn>1</mn></msub><mo><</mo><mi>⋯</mi><mo><</mo><msub><mi>i</mi> <mi>p</mi></msub><mo>≤</mo><mi>r</mi></mrow><annotation encoding="application/x-tex">1\leq i_0\lt i_1\lt\cdots\lt i_p\leq r</annotation></semantics></math>. Then the differential is given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></msub><mo>∧</mo><mi>⋯</mi><mo>∧</mo><msub><mi>e</mi> <mrow><msub><mi>i</mi> <mi>p</mi></msub></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow> <mi>p</mi></munderover><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><msub><mi>x</mi> <mrow><msub><mi>i</mi> <mi>k</mi></msub></mrow></msub><msub><mi>e</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></msub><mo>∧</mo><mi>⋯</mi><mo>∧</mo><msub><mover><mi>e</mi><mo stretchy="false">^</mo></mover> <mrow><msub><mi>i</mi> <mi>k</mi></msub></mrow></msub><mo>∧</mo><mi>⋯</mi><mo>∧</mo><msub><mi>e</mi> <mrow><msub><mi>i</mi> <mi>r</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex"> d(e_{i_0}\wedge \cdots \wedge e_{i_p}) = \sum_{k = 0}^{p}(-1)^{k+1} x_{i_k} e_{i_0}\wedge \cdots\wedge \hat{e}_{i_k} \wedge \cdots\wedge e_{i_r} </annotation></semantics></math></div> <p>The differential can be obtained from the faces of the obvious Koszul <a class="existingWikiWord" href="/nlab/show/semi-simplicial+object">semi-simplicial</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>-<a class="existingWikiWord" href="/nlab/show/module">module</a> and the cochain complex above is obtained by the usual alternating sum rule.</p> <p>Now let <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> be a finitely generated left <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>-module. Then the abelian <a class="existingWikiWord" href="/nlab/show/chain+homology">chain homology</a> <a class="existingWikiWord" href="/nlab/show/abelian+group">groups</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>q</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>r</mi></msub><mo>;</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>H</mi> <mi>q</mi></msub><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>r</mi></msub><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>R</mi></msub><mi>A</mi><mo stretchy="false">)</mo><mo>,</mo></mrow><annotation encoding="application/x-tex"> H_q(x_1,\ldots,x_r; A) = H_q(K(x_1,\ldots,x_r)\otimes_R A), </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>q</mi></msup><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>r</mi></msub><mo>;</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>H</mi> <mi>q</mi></msup><mo stretchy="false">(</mo><msub><mi>Hom</mi> <mi>R</mi></msub><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>r</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo></mrow><annotation encoding="application/x-tex"> H^q(x_1,\ldots,x_r;A) = H^q(Hom_R(K(x_1,\ldots,x_r),A)), </annotation></semantics></math></div> <p>together with <a class="existingWikiWord" href="/nlab/show/connecting+homomorphisms">connecting homomorphisms</a>, form a homological and cohomological <a class="existingWikiWord" href="/nlab/show/delta-functor">delta-functor</a> (in the sense of <a class="existingWikiWord" href="/nlab/show/Tohoku">Tohoku</a>) respectively, deriving the zero parts</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mn>0</mn></msub><mo>=</mo><mi>A</mi><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>r</mi></msub><mo stretchy="false">)</mo><mi>A</mi></mrow><annotation encoding="application/x-tex"> H_0 = A/(x_1,\ldots,x_r)A</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>0</mn></msup><mo>=</mo><msub><mi>Hom</mi> <mi>R</mi></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>r</mi></msub><mo stretchy="false">)</mo><mi>R</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H^0 = Hom_R(R/(x_1,\ldots,x_r)R,A)</annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>r</mi></msub><mo stretchy="false">)</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">(x_1,\ldots,x_r)A</annotation></semantics></math> is the left <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>-submodule generated by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>r</mi></msub></mrow><annotation encoding="application/x-tex">x_1,\ldots,x_r</annotation></semantics></math>. A Poincare-like duality holds: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>p</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>r</mi></msub><mo>;</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>H</mi> <mrow><mi>r</mi><mo>−</mo><mi>p</mi></mrow></msup><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>r</mi></msub><mo>;</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H_p(x_1,\ldots,x_r;A) = H^{r-p}(x_1,\ldots,x_r;A)</annotation></semantics></math>.</p> <p>The sequence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>x</mi></mstyle><mo>=</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>r</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{x} = (x_1,\ldots,x_r)</annotation></semantics></math> is called <strong><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>-regular</strong> (or regular on <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>) if 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> the image of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">x_i</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">A/(x_1,\ldots,x_{i-1})A</annotation></semantics></math> annihilates only zero. This terminology is in accord with calling a non-zero divisor in a ring a “regular element” (and is in accord with the terminology regular local rings).</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>x</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{x}</annotation></semantics></math> is a regular sequence on/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> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>x</mi></mstyle><mo>,</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(\mathbf{x},R)</annotation></semantics></math> is a free resolution of the module <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>r</mi></msub><mo stretchy="false">)</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">R/(x_1,\ldots,x_r)R</annotation></semantics></math> and the cohomology <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>q</mi></msup><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>r</mi></msub><mo>;</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>Ext</mi> <mi>R</mi> <mi>q</mi></msubsup><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>r</mi></msub><mo stretchy="false">)</mo><mi>R</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^q(x_1,\ldots,x_r;A) = Ext^q_R(R/(x_1,\ldots,x_r)R,A)</annotation></semantics></math> while <a class="existingWikiWord" href="/nlab/show/Koszul+homology">Koszul homology</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>q</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>r</mi></msub><mo>;</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>Tor</mi> <mi>q</mi> <mi>R</mi></msubsup><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>r</mi></msub><mo stretchy="false">)</mo><mi>R</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H_q(x_1,\ldots,x_r;A) = Tor_q^R(R/(x_1,\ldots,x_r)R,A)</annotation></semantics></math>.</p> <p>The resolution of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>r</mi></msub><mo stretchy="false">)</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">R/(x_1,\ldots,x_r)R</annotation></semantics></math> can be written</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><msup><mi>Λ</mi> <mi>r</mi></msup><mo stretchy="false">(</mo><msup><mi>R</mi> <mi>r</mi></msup><mo stretchy="false">)</mo><mo>→</mo><mi>⋯</mi><mo>→</mo><msup><mi>Λ</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><msup><mi>R</mi> <mi>r</mi></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mi>R</mi> <mi>r</mi></msup><mo>→</mo><mi>R</mi><mo>→</mo><mi>R</mi><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>r</mi></msub><mo stretchy="false">)</mo><mi>R</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0 \to \Lambda^r(R^r)\to \cdots \to \Lambda^2(R^r)\to R^r \to R \to R/(x_1,\ldots,x_r)R\to 0 </annotation></semantics></math></div> <p>and the <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>-linear map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>R</mi> <mi>r</mi></msup><mo>→</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">R^r\to R</annotation></semantics></math> is given by the row vector <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>r</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x_1,\ldots,x_r)</annotation></semantics></math>.</p> <h2 id="Properties">Properties</h2> <div class="num_prop" id="KoszulComplexOfRegularSequenceIsFreeResolutionOfQuotientRing"> <h6 id="proposition">Proposition</h6> <p><strong>(Koszul complex of <a class="existingWikiWord" href="/nlab/show/regular+sequence">regular sequence</a> is <a class="existingWikiWord" href="/nlab/show/free+resolution">free resolution</a> of <a class="existingWikiWord" href="/nlab/show/quotient+ring">quotient ring</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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>x</mi> <mi>d</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x_1, \cdots, x_d)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/regular+sequence">regular sequence</a> of elements 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>. Then the Koszul complex <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>d</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(x_1,\cdots, x_d)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/free+resolution">free resolution</a> of the <a class="existingWikiWord" href="/nlab/show/quotient+ring">quotient ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>x</mi> <mi>d</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R/(x_1, \cdots, x_d)</annotation></semantics></math>.</p> </div> <div class="num_prop" id="KoszulResolutionForNoetherianRngAndElementsInJacobson"> <h6 id="proposition_2">Proposition</h6> <p><strong>(Koszul resolution detected in degree (-1))</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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>x</mi> <mi>d</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x_1, \cdots, x_d)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/sequence">sequence</a> of elements 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>, such that</p> <ol> <li> <p><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> is <a class="existingWikiWord" href="/nlab/show/Noetherian+ring">Noetherian</a>;</p> </li> <li> <p>each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">x_i</annotation></semantics></math> is contained in the <a class="existingWikiWord" href="/nlab/show/Jacobson+radical">Jacobson radical</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></p> </li> </ol> <p>then the following are equivalent:</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a> of the Koszul complex <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>d</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(x_1, \cdots, x_d)</annotation></semantics></math> vanishes in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">-1</annotation></semantics></math>;</p> </li> <li> <p>the Koszul complex <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>d</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(x_1, \cdots, x_d)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/free+resolution">free resolution</a> of the <a class="existingWikiWord" href="/nlab/show/quotient+ring">quotient ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>x</mi> <mi>d</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R/(x_1, \cdots, x_d)</annotation></semantics></math>, hence its <a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a> vanishes in all degrees <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≤</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\leq -1</annotation></semantics></math>;</p> </li> <li> <p>the sequence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>x</mi> <mi>d</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x_1, \cdots, x_d)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/regular+sequence">regular sequence</a>.</p> </li> </ol> </div> <p>A <strong>proof</strong> is spelled out on <a href="https://en.wikipedia.org/wiki/Koszul_complex#Properties_of_a_Koszul_homology">Wikipedia - Properties of Koszul homology</a></p> <h2 id="examples">Examples</h2> <div class="num_example" id="KoszulComplexForFormalPowerSeriesAlgebras"> <h6 id="example">Example</h6> <p><strong>(Koszul complex for formal power series algebras)</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>R</mi><mo>=</mo><mi>k</mi><mo stretchy="false">[</mo><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">R = k[ [ X_1,\cdots, X_n ] ]</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/formal+power+series+algebra">formal power series algebra</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 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> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>f</mi> <mi>r</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f_1, \cdots, f_r)</annotation></semantics></math> be formal power series whose constant term vanishes. Then the Koszul complex is a homological resolution precisely already if its cohomology in degree -1 vanishes</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><msup><mi>H</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>f</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>⇔</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mi>K</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>f</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msub><mo>≃</mo> <mi>qi</mi></msub></mrow></mover><mi>k</mi><mo stretchy="false">[</mo><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>f</mi> <mi>r</mi></msub><mo stretchy="false">)</mo><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( H^{-1}(K(f_1, \cdots, f_n)) = 0 \right) \;\Leftrightarrow\; \left( K(f_1, \cdots, f_n) \overset{\simeq_{qi}}{\longrightarrow} k[ [X_1, \cdots X_n] ]/(f_1, \cdots, f_r) \right) \,. </annotation></semantics></math></div> <p>This is because the assumptions of prop. <a class="maruku-ref" href="#KoszulResolutionForNoetherianRngAndElementsInJacobson"></a> are met: A formal power series ring over a field is Noetherian (<a href="noetherian+ring#PolynomialAlgebraOverNoetherianRingIsNoetherian">this example</a>) and an element of a formal power series algebra is in the Jacobson radical precisely if its constant term vanishes (<a href="Jacobson+radical#JacobsonRadicalOfFormalPowerSeriesAlgebra">this example</a>).</p> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Koszul-Tate+resolution">Koszul-Tate resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/syzygy">syzygy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BV-formalism">BV-formalism</a></p> </li> </ul> <h2 id="references">References</h2> <p>The original reference is</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jean-Louis+Koszul">Jean-Louis Koszul</a>, <em>Homologie et cohomologie des algèbres de Lie</em> , Bulletin de la Société Mathématique de France, 78, 1950, pp 65-127.</li> </ul> <p>A standard textbook reference is</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Charles+Weibel">Charles Weibel</a>, section 4.5 of <em>Homological algebra</em> (<a href="http://www.math.unam.mx/javier/weibel.pdf">pdf</a>)</li> </ul> <p>A generalization of Koszul complexes to (appropriate resolutions of) <a class="existingWikiWord" href="/nlab/show/algebras+over+operads">algebras over operads</a> is in</p> <ul> <li>Joan Millès, <em>The Koszul complex is the cotangent complex</em>, MPIM2010-32, <a href="http://www.mpim-bonn.mpg.de/preprints/send?bid=4143">pdf</a></li> </ul> <p>See also</p> <ul> <li id="Wikipedia">Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Koszul_complex">Koszul complex</a></em></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on July 13, 2018 at 16:49:11. 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