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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> <blockquote> <p>under construction</p> </blockquote> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <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> <ul> <li><a href='#ContravariantExtOnObject'>Contravariant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ext</mi></mrow><annotation encoding="application/x-tex">Ext</annotation></semantics></math> on an ordinary object</a></li> <li><a href='#InTermsOfDerivedCategories'>In terms of derived categories</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#BasicProperties'>Basic properties</a></li> <li><a href='#RelationToGroupExtensions'>Relation to extensions</a></li> <ul> <li><a href='#1extensions_over_single_objects'>1-Extensions over single objects</a></li> <li><a href='#higher_extensions_over_general_chain_complexes'>Higher extensions over general chain complexes</a></li> </ul> <li><a href='#RelationToGroupCohomology'>Relation to group cohomology</a></li> <li><a href='#Locatization'>Localization</a></li> <li><a href='#techniques_for_constructing_'>Techniques for constructing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ext</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">Ext^n</annotation></semantics></math></a></li> <li><a href='#yoneda_product'>Yoneda product</a></li> </ul> <li><a href='#applications_in_cohomology'>Applications in cohomology</a></li> <ul> <li><a href='#universal_coefficient_theorem'>Universal coefficient theorem</a></li> <li><a href='#various_notions_of_cohomology_expressed_by_'>Various notions of cohomology expressed by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ext</mi></mrow><annotation encoding="application/x-tex">Ext</annotation></semantics></math></a></li> </ul> <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>In the context of <a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a> the <a class="existingWikiWord" href="/nlab/show/right+derived+functor">right derived functor</a> of the <a class="existingWikiWord" href="/nlab/show/hom-functor">hom-functor</a> is called the <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ext</mi></mrow><annotation encoding="application/x-tex">Ext</annotation></semantics></math>-functor</em> . It derives its name from the fact that the <a class="existingWikiWord" href="/nlab/show/derived+hom-functor">derived hom-functor</a> between <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a> classifies abelian <a class="existingWikiWord" href="/nlab/show/group+extensions">group extensions</a> of <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> by <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>. (This is a special case of the general classification of <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundles">principal ∞-bundles</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extensions">∞-group extensions</a> by general <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a>/<a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a>.)</p> <p>Together with the <a class="existingWikiWord" href="/nlab/show/Tor">Tor</a>-functor it is one of the central objects of interest in homological algebra.</p> <p>Given an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a> <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">\mathcal{A}</annotation></semantics></math> we may consider the <a class="existingWikiWord" href="/nlab/show/hom-functor">hom-functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>𝒜</mi></msub><mo>:</mo><msup><mi>𝒜</mi> <mi>op</mi></msup><mo>×</mo><mi>𝒜</mi><mo>→</mo></mrow><annotation encoding="application/x-tex">Hom_{\mathcal{A}} : \mathcal{A}^{op}\times \mathcal{A}\to </annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> either as a functor in first or in second argument, and compute the corresponding <a class="existingWikiWord" href="/nlab/show/right+derived+functors">right derived functors</a>.</p> <p>If they exist, the classical right derived functors of either functor agree and also agree with the <a class="existingWikiWord" href="/nlab/show/homology">homology</a> of the mixed <a class="existingWikiWord" href="/nlab/show/double+complex">double complex</a> obtained by taking simultaneously a <a class="existingWikiWord" href="/nlab/show/projective+resolution">projective resolution</a> of the first contravariant argument and an <a class="existingWikiWord" href="/nlab/show/injective+resolution">injective resolution</a> of the second covariant argument. The last construction is called the <em>balanced <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ext</mi></mrow><annotation encoding="application/x-tex">Ext</annotation></semantics></math>.</em></p> <p>Alternatively, one can consider the <a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(\mathcal{A})</annotation></semantics></math> and define</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ext</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≔</mo><msub><mi>Hom</mi> <mrow><mi>D</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">[</mo><mi>p</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Ext^p(X,A) \coloneqq Hom_{D(A)}(X,A[p]) </annotation></semantics></math></div> <p>or define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ext</mi> <mi>i</mi></msup></mrow><annotation encoding="application/x-tex">Ext^i</annotation></semantics></math>-groups as groups of abelian extensions of length <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>, discussed below at <em><a href="#RelationToGroupExtensions">Relation to extensions</a></em>.</p> <h2 id="definition">Definition</h2> <p>We give the definition following the discussion at <em><a class="existingWikiWord" href="/nlab/show/derived+functors+in+homological+algebra">derived functors in homological algebra</a></em>.</p> <h3 id="ContravariantExtOnObject">Contravariant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ext</mi></mrow><annotation encoding="application/x-tex">Ext</annotation></semantics></math> on an ordinary object</h3> <p>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">\mathcal{A}</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a> with <a class="existingWikiWord" href="/nlab/show/projective+object">enough projectives</a>. And let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">A \in \mathcal{A}</annotation></semantics></math> be any object. Consider the <a class="existingWikiWord" href="/nlab/show/contravariant+functor">contravariant</a> <a class="existingWikiWord" href="/nlab/show/hom-functor">hom-functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>𝒜</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>:</mo><msup><mi>𝒜</mi> <mi>op</mi></msup><mo>→</mo><mi>Ab</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Hom_{\mathcal{A}}(-, A) : \mathcal{A}^{op} \to Ab \,. </annotation></semantics></math></div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>This is a <a class="existingWikiWord" href="/nlab/show/left+exact+functor">left exact functor</a>.</p> <p>Therefore to derive it by <a class="existingWikiWord" href="/nlab/show/resolutions">resolutions</a> we need to consider <a class="existingWikiWord" href="/nlab/show/injective+resolutions">injective resolutions</a> in the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒜</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">\mathcal{A}^{op}</annotation></semantics></math>. But these are <a class="existingWikiWord" href="/nlab/show/projective+resolutions">projective resolutions</a> in <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">\mathcal{A}</annotation></semantics></math> itself.</p> </div> <div class="num_defn" id="OfSingleObjByProjResolution"> <h6 id="definition_2">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">X \in \mathcal{A}</annotation></semantics></math> any object and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>Q</mi><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi><mo stretchy="false">)</mo><mo>∈</mo><msub><mi>Ch</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">((Q X) \to X) \in Ch_{\bullet \geq 0}(\mathcal{A})</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/projective+resolution">projective resolution</a>, and for <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>, the <strong><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>th <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ext</mi></mrow><annotation encoding="application/x-tex">Ext</annotation></semantics></math>-group</strong> of <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> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in <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> is the degree-<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/cochain+cohomology">cochain cohomology</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ext</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≔</mo><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><msub><mi>Hom</mi> <mi>𝒜</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>Q</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Ext^n(X,A) \coloneqq H^n ( Hom_{\mathcal{A}}((Q X)_\bullet, A)) </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/cochain+complex">cochain complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>Q</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom((Q X)_\bullet, A)</annotation></semantics></math>.</p> </div> <p>The following proposition expands a bit on the meaning of this definition. Write</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><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo>:</mo><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>𝒜</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>×</mo><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>Ab</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> [-,-] : Ch_{\bullet}(\mathcal{A})^{op} \times Ch_\bullet(\mathcal{A}) \to Ch_\bullet(Ab) </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/internal+hom+of+chain+complexes">enriched hom of chain complexes</a>.</p> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>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>th Ext-group is canonically identified with the 0-th <a class="existingWikiWord" href="/nlab/show/homology">homology</a> of this enriched hom from the resolution <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">Q X</annotation></semantics></math> of <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> to 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/delooping">delooping</a>/<a class="existingWikiWord" href="/nlab/show/suspension">suspension</a> chain complex of <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> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>A</mi><mo>=</mo><mi>A</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}^n A = A[n]</annotation></semantics></math> (concentrated 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> in degree <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>):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ext</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>H</mi> <mn>0</mn></msub><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>Q</mi><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>A</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>;</mo></mrow><annotation encoding="application/x-tex"> Ext^n(X,A) \simeq H_0 [(Q X), A[n] ] \,; </annotation></semantics></math></div> <p>or equivalently, if we think of degree <a class="existingWikiWord" href="/nlab/show/chain+homology">chain homology</a> as the 0th <a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a> (under <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a>) and write 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/suspension">suspension</a>/<a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> of <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> as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>A</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}^n A</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ext</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>Q</mi><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>A</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Ext^n(X,A) \simeq \pi_0 [(Q X), \mathbf{B}^n A ] \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof">Proof</h6> <p>This is a special case of the general discussion at <a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a>.</p> <p>By the discussion at <em><a class="existingWikiWord" href="/nlab/show/internal+hom+of+chain+complexes">internal hom of chain complexes</a></em>, the 0-<a class="existingWikiWord" href="/nlab/show/cycles">cycles</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>Q</mi><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>A</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[(Q X), \mathbf{B}^n A ]</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/chain+maps">chain maps</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><mrow><mtable><mtr><mtd><mi>⋮</mi></mtd> <mtd></mtd> <mtd><mi>⋮</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>Q</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></mover></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msubsup><mo>∂</mo> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow> <mrow><mi>Q</mi><mi>X</mi></mrow></msubsup></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>Q</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>f</mi> <mi>n</mi></msub></mrow></mover></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msubsup><mo>∂</mo> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow> <mrow><mi>Q</mi><mi>X</mi></mrow></msubsup></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>Q</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></mover></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>⋮</mi></mtd> <mtd></mtd> <mtd><mi>⋮</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msubsup><mo>∂</mo> <mn>1</mn> <mrow><mi>Q</mi><mi>X</mi></mrow></msubsup></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>Q</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mn>1</mn></msub></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msubsup><mo>∂</mo> <mn>0</mn> <mrow><mi>Q</mi><mi>X</mi></mrow></msubsup></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>Q</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mn>0</mn></msub></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>f</mi> <mn>0</mn></msub></mrow></mover></mtd> <mtd><mn>0</mn></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \vdots && \vdots \\ \downarrow && \downarrow \\ (Q X)_{n+1} &\stackrel{f_{n+1}}{\to}& 0 \\ \downarrow^{\mathrlap{\partial^{Q X}_{n-1}}} && \downarrow \\ (Q X)_{n} &\stackrel{f_{n}}{\to}& A \\ \downarrow^{\mathrlap{\partial^{Q X}_{n-1}}} && \downarrow \\ (Q X)_{n-1} &\stackrel{f_{n-1}}{\to}& 0 \\ \downarrow && \downarrow \\ \vdots && \vdots \\ \downarrow^{\mathrlap{\partial^{Q X}_1}} && \downarrow \\ (Q X)_1 &\stackrel{f_1}{\to}& 0 \\ \downarrow^{\mathrlap{\partial^{Q X}_0}} && \downarrow \\ (Q X)_0 &\stackrel{f_0}{\to}& 0 } \,. </annotation></semantics></math></div> <p>By the definition of chain maps this are precisely those morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>n</mi></msub><mo>:</mo><mo stretchy="false">(</mo><mi>Q</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">f_n : (Q X)_n \to A</annotation></semantics></math> such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>d</mi> <mi>n</mi></msup><msub><mi>f</mi> <mi>n</mi></msub><mo>≔</mo><msub><mi>f</mi> <mi>n</mi></msub><mo>∘</mo><msubsup><mo>∂</mo> <mi>n</mi> <mrow><mi>Q</mi><mi>X</mi></mrow></msubsup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> d^n f_n \coloneqq f_n \circ \partial^{Q X}_n = 0 </annotation></semantics></math></div> <p>which exhibits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">f_n</annotation></semantics></math> as a degree-<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/cochain">cochain</a> in the <a class="existingWikiWord" href="/nlab/show/cochain+complex">cochain complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>Q</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom((Q X)_\bullet, A)</annotation></semantics></math>.</p> <p>Similarly, the <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>-<a class="existingWikiWord" href="/nlab/show/boundaries">boundaries</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>Q</mi><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>A</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[(Q X), \mathbf{B}^n A]</annotation></semantics></math> come from <a class="existingWikiWord" href="/nlab/show/chain+homotopies">chain homotopies</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo>:</mo><mn>0</mn><mo>⇒</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">\lambda : 0 \Rightarrow 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><mrow><mtable><mtr><mtd><mi>⋮</mi></mtd> <mtd></mtd> <mtd><mi>⋮</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>Q</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></mover></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msubsup><mo>∂</mo> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow> <mrow><mi>Q</mi><mi>X</mi></mrow></msubsup></mrow></mpadded></msup></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mrow><msub><mi>λ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></mpadded></msub></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>Q</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>f</mi> <mi>n</mi></msub></mrow></mover></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msubsup><mo>∂</mo> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow> <mrow><mi>Q</mi><mi>X</mi></mrow></msubsup></mrow></mpadded></msup></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mrow><msub><mi>λ</mi> <mi>n</mi></msub></mrow></mpadded></msub></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>Q</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></mover></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>⋮</mi></mtd> <mtd></mtd> <mtd><mi>⋮</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msubsup><mo>∂</mo> <mn>1</mn> <mrow><mi>Q</mi><mi>X</mi></mrow></msubsup></mrow></mpadded></msup></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mrow><msub><mi>λ</mi> <mn>2</mn></msub></mrow></mpadded></msub></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>Q</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mn>1</mn></msub></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msubsup><mo>∂</mo> <mn>0</mn> <mrow><mi>Q</mi><mi>X</mi></mrow></msubsup></mrow></mpadded></msup></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mrow><msub><mi>λ</mi> <mn>1</mn></msub></mrow></mpadded></msub></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>Q</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mn>0</mn></msub></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>f</mi> <mn>0</mn></msub></mrow></mover></mtd> <mtd><mn>0</mn></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \vdots && \vdots \\ \downarrow && \downarrow \\ (Q X)_{n+1} &\stackrel{f_{n+1}}{\to}& 0 \\ \downarrow^{\mathrlap{\partial^{Q X}_{n-1}}} &\nearrow_{\mathrlap{\lambda_{n+1}}}& \downarrow \\ (Q X)_{n} &\stackrel{f_{n}}{\to}& A \\ \downarrow^{\mathrlap{\partial^{Q X}_{n-1}}} &\nearrow_{\mathrlap{\lambda_{n}}}& \downarrow \\ (Q X)_{n-1} &\stackrel{f_{n-1}}{\to}& 0 \\ \downarrow && \downarrow \\ \vdots && \vdots \\ \downarrow^{\mathrlap{\partial^{Q X}_1}} &\nearrow_{\mathrlap{\lambda_{2}}}& \downarrow \\ (Q X)_1 &\stackrel{f_1}{\to}& 0 \\ \downarrow^{\mathrlap{\partial^{Q X}_0}} &\nearrow_{\mathrlap{\lambda_{1}}}& \downarrow \\ (Q X)_0 &\stackrel{f_0}{\to}& 0 } \,. </annotation></semantics></math></div> <p>in that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>n</mi></msub><mo>=</mo><msub><mi>λ</mi> <mi>n</mi></msub><mo>∘</mo><msubsup><mo>∂</mo> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow> <mrow><mi>Q</mi><mi>X</mi></mrow></msubsup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f_n = \lambda_n \circ \partial^{Q X}_{n-1} \,. </annotation></semantics></math></div> <p>This are precisely the degree-<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/coboundaries">coboundaries</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>Q</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom((Q X)_\bullet, A)</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>This perspective on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ext</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">Ext^n</annotation></semantics></math>-group as being the <a class="existingWikiWord" href="/nlab/show/homotopy+classes">homotopy classes</a> of maps out of (a resolution of) <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> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>A</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}^n A</annotation></semantics></math> is made more manifest in the discussion <a href="#InTermsOfDerivedCategories">in terms of derived categories</a> below. It connects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ext</mi></mrow><annotation encoding="application/x-tex">Ext</annotation></semantics></math>-groups and their <a href="#RelationToGroupExtensions">relation to extensions</a> to the general context of <em><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></em> and <em><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extensions">∞-group extensions</a></em>. See at <em><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></em> for more on this.</p> </div> <h3 id="InTermsOfDerivedCategories">In terms of derived categories</h3> <p>(…)</p> <p>(<a href="#KashiwaraShapira">Kashiwara-Shapira</a>)</p> <p>(…)</p> <h2 id="properties">Properties</h2> <h3 id="BasicProperties">Basic properties</h3> <p> <div class='num_prop' id='ExtCompatibleWithDirectSumsAndProducts'> <h6>Proposition</h6> <p></p> <p>For all <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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>∈</mo><mi>Rings</mi></mrow><annotation encoding="application/x-tex">R \in Rings</annotation></semantics></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ext</mi> <mi>R</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ext^n_R(-,-)</annotation></semantics></math>-functor sends <a class="existingWikiWord" href="/nlab/show/direct+sums">direct sums</a> in the first variable and <a class="existingWikiWord" href="/nlab/show/direct+products">direct products</a> in the second variable to direct products:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ext</mi> <mi>R</mi> <mi>n</mi></msubsup><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mo>⊕</mo> <mi>i</mi></msub><msub><mi>A</mi> <mi>i</mi></msub><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><mi>B</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mi>i</mi></munder><msubsup><mi>Ext</mi> <mi>R</mi> <mi>n</mi></msubsup><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>A</mi> <mi>i</mi></msub><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><mi>B</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex"> Ext^n_R \big( \oplus_i A_i \,,\, B \big) \;\simeq\; \prod_i Ext^n_R \big( A_i \,,\, B \big) </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><msubsup><mi>Ext</mi> <mi>R</mi> <mi>n</mi></msubsup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>A</mi><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mi>i</mi></munder><msub><mi>B</mi> <mi>i</mi></msub><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mi>i</mi></munder><msubsup><mi>Ext</mi> <mi>R</mi> <mi>n</mi></msubsup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>A</mi><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>B</mi> <mi>i</mi></msub><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Ext^n_R \big( A \,,\, \prod_i B_i \big) \;\simeq\; \prod_i Ext^n_R \big( A \,,\, B_i \big) \,. </annotation></semantics></math></div> <p></p> </div> </p> <p>(e.g. <a href="#Weibel">Weibel, Prop. 3.3.4</a>)</p> <p>In particular, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ext</mi> <mi>R</mi> <mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">Ext^n_R</annotation></semantics></math> preserves the <a class="existingWikiWord" href="/nlab/show/finite+colimit">finite</a> <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> (a <a class="existingWikiWord" href="/nlab/show/biproduct">biproduct</a>) in both variables:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ext</mi> <mi>R</mi> <mi>n</mi></msubsup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>A</mi><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>B</mi> <mn>1</mn></msub><mo>⊕</mo><msub><mi>B</mi> <mn>2</mn></msub><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msubsup><mi>Ext</mi> <mi>R</mi> <mi>n</mi></msubsup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>A</mi><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>B</mi> <mn>1</mn></msub><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>⊕</mo><msubsup><mi>Ext</mi> <mi>R</mi> <mi>n</mi></msubsup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>A</mi><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>B</mi> <mn>2</mn></msub><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Ext^n_R \big( A \,,\, B_1 \oplus B_2 \big) \;\simeq\; Ext^n_R \big( A \,,\, B_1 \big) \oplus Ext^n_R \big( A \,,\, B_2 \big) \,. </annotation></semantics></math></div> <h3 id="RelationToGroupExtensions">Relation to extensions</h3> <p>We discuss how the group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ext</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ext^n(X,A)</annotation></semantics></math> is identified with the group of <a class="existingWikiWord" href="/nlab/show/extensions">extensions</a> of <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> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>A</mi><mo>=</mo><mi>A</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}^{n-1} A = A[n-1]</annotation></semantics></math>. In particular for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n = 1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathcal{A} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> this means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ext</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ext^1(X,A)</annotation></semantics></math> classified ordinary <a class="existingWikiWord" href="/nlab/show/group+extensions">group extensions</a> of <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> by <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>.</p> <p>This is the relation that the name “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ext</mi></mrow><annotation encoding="application/x-tex">Ext</annotation></semantics></math>” derives from. At <em><a class="existingWikiWord" href="/nlab/show/infinity-group+extension">infinity-group extension</a></em> is discussed how this relation is a special case of the more general relation that identifies <a class="existingWikiWord" href="/nlab/show/derived+hom-spaces">derived hom-spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}(X,\mathbf{B}^{n+1} A)</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>A</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}^n A</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundles">principal ∞-bundles</a> over <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>.</p> <h4 id="1extensions_over_single_objects">1-Extensions over single objects</h4> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>A</mi><mo>∈</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">X,A \in \mathcal{A}</annotation></semantics></math> two objects, an <strong><a class="existingWikiWord" href="/nlab/show/extension">extension</a></strong> of <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> by <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> is a <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a></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><mi>A</mi><mo>→</mo><mi>P</mi><mo>→</mo><mi>X</mi><mo>→</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> 0 \to A \to P \to X \to 0 \,. </annotation></semantics></math></div> <p>An <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of two such extensions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>P</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">P_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>P</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">P_2</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>P</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>P</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">P_1 \to P_2</annotation></semantics></math> in <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">\mathcal{A}</annotation></semantics></math> fitting into a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</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><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>P</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↗</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>P</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && P_1 \\ & \nearrow & & \searrow \\ A && \downarrow && X \\ & \searrow & & \nearrow \\ && P_2 } \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>All these homomophisms are necessarily <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a>, by the <a class="existingWikiWord" href="/nlab/show/short+five+lemma">short five lemma</a>.</p> </div> <div class="num_defn"> <h6 id="definition_4">Definition</h6> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ext</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ext(X,A)</annotation></semantics></math> for the set of <a class="existingWikiWord" href="/nlab/show/isomorphism+classes">isomorphism classes</a> of such extensions.</p> </div> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>Under <a class="existingWikiWord" href="/nlab/show/Baer+sum">Baer sum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ext</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ext(X,A)</annotation></semantics></math> becomes an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>.</p> </div> <div class="num_defn" id="ExtractCocycleFromExtension"> <h6 id="definition_5">Definition</h6> <p>Define a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ExtractCocycle</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Extensions</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⟶</mo><msup><mi>Ext</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> ExtractCocycle \;\colon\; Extensions(X,A) \longrightarrow Ext^1(X,A) </annotation></semantics></math></div> <p>by the following construction:</p> <ol> <li> <p>Choose a <a class="existingWikiWord" href="/nlab/show/projective+presentation">projective presentation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>↪</mo><mi>Q</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">N \hookrightarrow Q \to X</annotation></semantics></math> of <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>. Then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A \to P \to X</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/algebra+extension">extension</a> consider the <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a></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>N</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Q</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mrow><mi>σ</mi><msub><mo stretchy="false">|</mo> <mi>N</mi></msub></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mi>σ</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mo>=</mo></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>P</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ N &\longrightarrow& Q &\longrightarrow& X \\ \big\downarrow {}^{\mathrlap{\sigma|_N}} && \big\downarrow {}^{\mathrlap{\sigma}} && \big\downarrow {}^{\mathrlap{=}} \\ A &\longrightarrow& P &\longrightarrow& X } \,, </annotation></semantics></math></div> <p>where</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> is any choice of <a class="existingWikiWord" href="/nlab/show/lift">lift</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Q \to X</annotation></semantics></math> through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math>, which exists by definition since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/projective+object">projective object</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><msub><mo stretchy="false">|</mo> <mi>N</mi></msub></mrow><annotation encoding="application/x-tex">\sigma|_N</annotation></semantics></math> is the induced morphism on the <a class="existingWikiWord" href="/nlab/show/fibers">fibers</a>, which exists by the <a class="existingWikiWord" href="/nlab/show/exact+sequence">exactness</a> of the two sequences.</p> </li> </ul> </li> <li> <p>By prop. <a class="maruku-ref" href="#Ext1FromProjectivePresentation"></a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><msub><mo stretchy="false">|</mo> <mi>N</mi></msub></mrow><annotation encoding="application/x-tex">\sigma|_N</annotation></semantics></math> represents an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">[</mo><mi>σ</mi><msub><mo stretchy="false">|</mo> <mi>N</mi></msub><mo maxsize="1.2em" minsize="1.2em">]</mo><mo>∈</mo><msup><mi>Ext</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \big[ \sigma|_N \big] \in Ext^1(X,A)</annotation></semantics></math>. Let this be the <a class="existingWikiWord" href="/nlab/show/image">image</a> of the map to be defined:</p> </li> </ol> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ExtractCocycle</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mi>A</mi><mo>→</mo><mi>P</mi><mo>→</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>↦</mo><mspace width="thickmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">[</mo><mi>σ</mi><msub><mo stretchy="false">|</mo> <mi>N</mi></msub><mo maxsize="1.2em" minsize="1.2em">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> ExtractCocycle \;\colon\; (A \to P \to X) \;\mapsto\; \big[ \sigma|_N \big] \,. </annotation></semantics></math></div> <p>This construction is independent of the choice of <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> involved and hence the map is well-defined.</p> </div> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>The map from def. <a class="maruku-ref" href="#ExtractCocycleFromExtension"></a> is a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a> of abelian groups</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ext</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><msup><mi>Ext</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Ext(X,A) \stackrel{\simeq}{\longrightarrow} Ext^1(X,A) \,. </annotation></semantics></math></div></div> <h4 id="higher_extensions_over_general_chain_complexes">Higher extensions over general chain complexes</h4> <p>(…)</p> <p>Given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>g</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>ℝ</mi><mi>Hom</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[g] \in \mathbb{R}Hom(X, A[n])</annotation></semantics></math>.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Q \to X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/projective+resolution">projective resolution</a>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mi>Q</mi><mo>→</mo><mi>A</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">g : Q \to A[n]</annotation></semantics></math> be a representative of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>g</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[g]</annotation></semantics></math>.<br />Consider the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a></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>P</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>cone</mi><mo stretchy="false">(</mo><mn>0</mn><mo>→</mo><mi>A</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Q</mi></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><mi>A</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ P &\to& cone(0 \to A[n]) \\ \downarrow && \downarrow \\ Q &\stackrel{g}{\to}& A[n] \\ \downarrow \\ X } </annotation></semantics></math></div> <p>(…)</p> <h3 id="RelationToGroupCohomology">Relation to group cohomology</h3> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/discrete+group">discrete group</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">[</mo><mi>G</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}[G]</annotation></semantics></math> its <a class="existingWikiWord" href="/nlab/show/group+ring">group ring</a>, over the <a class="existingWikiWord" href="/nlab/show/integers">integers</a>, and for <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 linear <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/representation">representation</a>, hence a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">[</mo><mi>G</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}[G]</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/module">module</a>, the <a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</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> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ext</mi> <mrow><mi>ℤ</mi><mo stretchy="false">[</mo><mi>G</mi><mo stretchy="false">]</mo><mi>Mod</mi></mrow> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>ℤ</mi><mo>,</mo><mi>N</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Ext^\bullet_{\mathbb{Z}[G]Mod}(\mathbb{Z}, N) \,. </annotation></semantics></math></div> <p>(…)</p> <h3 id="Locatization">Localization</h3> <p>(…)</p> <h3 id="techniques_for_constructing_">Techniques for constructing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ext</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">Ext^n</annotation></semantics></math></h3> <p>We discuss some facts helpful for the construction of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ext</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">Ext^n</annotation></semantics></math>-groups in certain situations.</p> <div class="num_prop"> <h6 id="proposition_4">Proposition</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">X \in \mathcal{A}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/projective+object">projective object</a>, then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ext</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> Ext^n(X, -) = 0 </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/zero+object">zero</a>-<a class="existingWikiWord" href="/nlab/show/functor">functor</a> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n \geq 1</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>The covariant <a class="existingWikiWord" href="/nlab/show/hom-functor">hom-functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom(X,-)</annotation></semantics></math> is generally a <a class="existingWikiWord" href="/nlab/show/left+exact+functor">left exact functor</a>. By the construction of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ext</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">Ext^n</annotation></semantics></math> via <a class="existingWikiWord" href="/nlab/show/projective+resolutions">projective resolutions</a>, def. <a class="maruku-ref" href="#OfSingleObjByProjResolution"></a>, it is sufficient to show that it is also a <a class="existingWikiWord" href="/nlab/show/right+exact+functor">right exact functor</a> if <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 projective. In fact, this is one of the equivalent characterizations of <em><a class="existingWikiWord" href="/nlab/show/projective+objects">projective objects</a></em> (ee the section <a href="projective+object#EquivalentCharacterizationInAbelianCats">projective object – in abelian categories – equivalent characterizations</a> for details).</p> </div> <div class="num_prop" id="Ext1FromProjectivePresentation"> <h6 id="proposition_5">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>A</mi><mo>∈</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">X, A \in \mathcal{A}</annotation></semantics></math> two objects, and</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><mi>N</mi><mover><mo>↪</mo><mi>i</mi></mover><mi>P</mi><mover><mo>→</mo><mi>p</mi></mover><mi>X</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0 \to N \stackrel{i}{\hookrightarrow} P \stackrel{p}{\to} X \to 0 </annotation></semantics></math></div> <p>a <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a> with <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> a <a class="existingWikiWord" href="/nlab/show/projective+object">projective object</a>, hence exhibiting a <a class="existingWikiWord" href="/nlab/show/projective+presentation">projective presentation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>≃</mo><mi>coker</mi><mo stretchy="false">(</mo><mi>N</mi><mo>↪</mo><mi>P</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \simeq coker(N \hookrightarrow P)</annotation></semantics></math> of <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>, there is an <a class="existingWikiWord" href="/nlab/show/exact+sequence">exact sequence</a></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><mi>Hom</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></mover><mi>Hom</mi><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></mover><mi>Hom</mi><mo stretchy="false">(</mo><mi>N</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>Ext</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0 \to Hom(X,A) \stackrel{Hom(p,A)}{\to} Hom(P, A) \stackrel{Hom(i,A)}{\to} Hom(N,A) \to Ext^1(X,A) \to 0 </annotation></semantics></math></div> <p>exhibiting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ext</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ext^1(X,A)</annotation></semantics></math> as the <a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom(i,A)</annotation></semantics></math>.</p> </div> <h3 id="yoneda_product">Yoneda product</h3> <p>The <span class="newWikiWord">Yoneda product<a href="/nlab/new/Yoneda+product">?</a></span> is a pairing</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ext</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>M</mi><mo stretchy="false">)</mo><mo>⊗</mo><msup><mi>Ext</mi> <mi>m</mi></msup><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>N</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>Ext</mi> <mrow><mi>n</mi><mo>+</mo><mi>m</mi></mrow></msup><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>M</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>N</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> Ext^n(A,M) \otimes Ext^m(A,N) \to Ext^{n+m}(A,M\otimes_A N). </annotation></semantics></math></div> <p>(…)</p> <h2 id="applications_in_cohomology">Applications in cohomology</h2> <p>A <a class="existingWikiWord" href="/nlab/show/derived+hom-functor">derived hom-functor</a> such as the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ext</mi></mrow><annotation encoding="application/x-tex">Ext</annotation></semantics></math> on chain complexes compute general notions of <em><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></em> (see the discussion there). Here we list some specific incarnations of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ext</mi></mrow><annotation encoding="application/x-tex">Ext</annotation></semantics></math>-construction in the context of cohomology.</p> <h3 id="universal_coefficient_theorem">Universal coefficient theorem</h3> <p>The <em><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></em> identifies, under suitable conditions, <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> to the <a class="existingWikiWord" href="/nlab/show/duality">dual</a> of <a class="existingWikiWord" href="/nlab/show/homology">homology</a> up to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ext</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">Ext^1</annotation></semantics></math>-groups.</p> <h3 id="various_notions_of_cohomology_expressed_by_">Various notions of cohomology expressed by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ext</mi></mrow><annotation encoding="application/x-tex">Ext</annotation></semantics></math></h3> <p>Various notions of <a class="existingWikiWord" href="/nlab/show/cohomology+groups">cohomology groups</a> in the context of <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a> can be expressed as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ext</mi></mrow><annotation encoding="application/x-tex">Ext</annotation></semantics></math>-groups, for instance:</p> <ul> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/discrete+group">discrete group</a>, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">[</mo><mi>G</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">K[G]</annotation></semantics></math> its <a class="existingWikiWord" href="/nlab/show/group+ring">group ring</a> over a commutative ring K, and for <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 <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>-linear <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/representation">representation</a> (hence a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">[</mo><mi>G</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">K[G]</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/module">module</a>), the <a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</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> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ext</mi> <mrow><mi>K</mi><mo stretchy="false">[</mo><mi>G</mi><mo stretchy="false">]</mo><mi>Mod</mi></mrow> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>K</mi><mo>,</mo><mi>N</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Ext^\bullet_{K[G]Mod}(K, N) \,. </annotation></semantics></math></div></li> <li> <p>For <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> an <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a> over some <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> and <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> an <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>-<a class="existingWikiWord" href="/nlab/show/bimodule">bimodule</a>, hence an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊗</mo><msup><mi>A</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">A \otimes A^{op}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/module">module</a>,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ext</mi> <mrow><mo stretchy="false">(</mo><mi>A</mi><mo>⊗</mo><msup><mi>A</mi> <mi>op</mi></msup><mo stretchy="false">)</mo><mi>Mod</mi></mrow> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>N</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Ext^\bullet_{(A \otimes A^{op})Mod}(A, N) </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a> of <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> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</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>.</p> </li> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a> over a commutative ring <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 <a class="existingWikiWord" href="/nlab/show/universal+enveloping+algebra">universal enveloping algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒰</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{U}(\mathfrak{g})</annotation></semantics></math>, and <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 Lie algebra module (hence an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒰</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{U}(\mathfrak{g})</annotation></semantics></math>-module), the <a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</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> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ext</mi> <mrow><mi>𝒰</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mi>Mod</mi></mrow> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>K</mi><mo>,</mo><mi>N</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Ext^\bullet_{\mathcal{U}(\mathfrak{g}) Mod}(K, N) \,. </annotation></semantics></math></div></li> </ul> <h2 id="Examples">Examples</h2> <div class="num_prop" id="ExtensionsOfTheIntegersAreTrivial"> <h6 id="proposition_6">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/group+extensions">group extensions</a> of the <a class="existingWikiWord" href="/nlab/show/integers">integers</a> are <a class="existingWikiWord" href="/nlab/show/trivial+group">trivial</a>)</strong></p> <p>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ext</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">Ext^1</annotation></semantics></math>-group of the <a class="existingWikiWord" href="/nlab/show/integers">integers</a> with coefficients in any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">A \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/AbelianGroups">AbelianGroups</a> is <a class="existingWikiWord" href="/nlab/show/trivial+group">trivial</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ext</mi> <mn>1</mn></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>ℤ</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Ext^1\big( \mathbb{Z}, \, A \big) \;\simeq\; 0 \,. </annotation></semantics></math></div></div> <p>(since the integers are already projective, e.g. <a href="#Boardman">Boardman, Prop. 19</a>)</p> <div class="num_prop" id="GroupExtensionsOfFiniteCyclicGroups"> <h6 id="proposition_7">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/group+extensions">group extensions</a> of <a class="existingWikiWord" href="/nlab/show/finite+group">finite</a> <a class="existingWikiWord" href="/nlab/show/cyclic+groups">cyclic groups</a>)</strong></p> <p>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ext</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">Ext^1</annotation></semantics></math>-group of the <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a> of <a class="existingWikiWord" href="/nlab/show/order+of+a+group">order</a> <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> with coefficients in any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">A \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/AbelianGroups">AbelianGroups</a> is the <a class="existingWikiWord" href="/nlab/show/quotient+group">quotient group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">/</mo><mi>n</mi><mi>A</mi><mo>≔</mo><mi>coker</mi><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>A</mi><mover><mo>⟶</mo><mrow><mi>n</mi><mo>⋅</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mover><mi>A</mi><mo maxsize="1.8em" minsize="1.8em">)</mo></mrow><annotation encoding="application/x-tex">A/n A \coloneqq coker\Big( A \overset{n \cdot(-)}{\longrightarrow} A \Big)</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ext</mi> <mn>1</mn></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mi>n</mi><mi>ℤ</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>A</mi><mo stretchy="false">/</mo><mi>n</mi><mi>A</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Ext^1 \big( \mathbb{Z} / n\mathbb{Z}, \, A \big) \;\simeq\; A / n A \,. </annotation></semantics></math></div></div> <p>(e.g. <a href="#Boardman">Boardman, Prop. 20</a>)</p> <p>But:</p> <p> <div class='num_prop'> <h6>Proposition</h6> <p></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ext</mi> <mrow><mo>≥</mo><mn>2</mn></mrow></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mi>n</mi><mi>ℤ</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Ext^{\geq 2} \big( \mathbb{Z} / n\mathbb{Z}, \, A \big) \;\simeq\; 0 \,. </annotation></semantics></math></div> <p></p> </div> </p> <p>(e.g. <a href="https://www.math.purdue.edu/~arapura/algebra/homological3.pdf#page=8">here</a>)</p> <div class="num_example" id="ExamplesOfExtGroupsOfFiniteCyclicGroups"> <h6 id="examples_2">Examples</h6> <p>As special cases of Prop. <a class="maruku-ref" href="#GroupExtensionsOfFiniteCyclicGroups"></a> we have:</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ext</mi> <mn>1</mn></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mi>n</mi><mi>ℤ</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>ℤ</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>ℤ</mi><mo stretchy="false">/</mo><mi>n</mi><mi>ℤ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Ext^1 \big( \mathbb{Z} / n\mathbb{Z}, \, \mathbb{Z} \big) \;\simeq\; \mathbb{Z} / n\mathbb{Z} \,. </annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ext</mi> <mn>1</mn></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mi>n</mi><mi>ℤ</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>ℤ</mi><mo stretchy="false">/</mo><mi>m</mi><mi>ℤ</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>ℤ</mi><mo stretchy="false">/</mo><mi>d</mi><mi>ℤ</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> Ext^1 \big( \mathbb{Z} / n\mathbb{Z}, \, \mathbb{Z} / m\mathbb{Z} \big) \;\simeq\; \mathbb{Z} / d\mathbb{Z} \,, </annotation></semantics></math></p> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>≔</mo></mrow><annotation encoding="application/x-tex">d \coloneqq</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/gcd">gcd</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n,m)</annotation></semantics></math> .</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ext</mi> <mn>1</mn></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mi>n</mi><mi>ℤ</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>ℚ</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Ext^1 \big( \mathbb{Z} / n\mathbb{Z}, \, \mathbb{Q} \big) \;\simeq\; 0 \,. </annotation></semantics></math></p> </li> </ul> </div> <p>(e.g. <a href="#Boardman">Boardman, Cor. 21</a>)</p> <p>In fact the last case of Example <a class="maruku-ref" href="#ExamplesOfExtGroupsOfFiniteCyclicGroups"></a> generalizes beyond cyclic groups:</p> <div class="num_prop" id="ExtensionsByRationalNumebrsAreTrivial"> <h6 id="proposition_8">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a> by the <a class="existingWikiWord" href="/nlab/show/rational+numbers">rational numbers</a> are trivial)</strong></p> <p>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ext</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">Ext^1</annotation></semantics></math>-group of any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">A \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/AbelianGroups">AbelianGroups</a> with coefficients in the <a class="existingWikiWord" href="/nlab/show/rational+numbers">rational numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}</annotation></semantics></math> is trivial:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ext</mi> <mn>1</mn></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>A</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>ℚ</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Ext^1\big( A, \, \mathbb{Q} \big) \;\simeq\; 0 \,. </annotation></semantics></math></div></div> <p>(e.g. <a href="#Boardman">Boardman, Prop. 22</a>)</p> <p>Less immediate is this example:</p> <div class="num_prop" id="ExtensionsByRationalNumebrsAreTrivial"> <h6 id="proposition_9">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a> of <a class="existingWikiWord" href="/nlab/show/rational+numbers">rational numbers</a> by the <a class="existingWikiWord" href="/nlab/show/integers">integers</a>)</strong></p> <p>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ext</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">Ext^1</annotation></semantics></math>-group of the <a class="existingWikiWord" href="/nlab/show/rational+numbers">rational numbers</a> by the <a class="existingWikiWord" href="/nlab/show/integers">integers</a> is the <a class="existingWikiWord" href="/nlab/show/quotient+group">quotient group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔸</mi><mo stretchy="false">/</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{A}/\mathbb{Q}</annotation></semantics></math></p> <ul> <li> <p>of the <a class="existingWikiWord" href="/nlab/show/ring+of+adeles">group of adeles</a> (without the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a>-factor), i.e. the <a class="existingWikiWord" href="/nlab/show/restricted+product">restricted product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔸</mi><mspace width="thinmathspace"></mspace><mo>≔</mo><mspace width="thinmathspace"></mspace><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mrow><mi>p</mi><mspace width="thickmathspace"></mspace><mi>prime</mi></mrow><mstyle scriptlevel="0"><mo>′</mo></mstyle></munderover><msub><mi>ℚ</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{A} \,\coloneqq\, \underoverset{p\;prime}{\prime}{\prod} \mathbb{Q}_p</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℚ</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{Q}_p</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/p-adic+numbers">p-adic numbers</a> restricted along the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mi>p</mi></msub><mo>→</mo><msub><mi>ℚ</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_p \to \mathbb{Q}_p</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/p-adic+integers">p-adic integers</a>;</p> </li> <li> <p>by the <a class="existingWikiWord" href="/nlab/show/rational+numbers">rational numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}</annotation></semantics></math> (…):</p> </li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ext</mi> <mn>1</mn></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>ℚ</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>ℤ</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>𝔸</mi><mo stretchy="false">/</mo><mi>ℚ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Ext^1\big( \mathbb{Q}, \, \mathbb{Z} \big) \;\simeq\; \mathbb{A}/\mathbb{Q} \,. </annotation></semantics></math></div></div> <p>(<a href="#Boardman">Boardman, Theorem 25</a>)</p> <h2 id="related_concepts">Related concepts</h2> <div> <table><thead><tr><th></th><th><a class="existingWikiWord" href="/nlab/show/homotopy+%28as+an+operation%29">homotopy</a></th><th><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></th><th><a class="existingWikiWord" href="/nlab/show/homology">homology</a></th></tr></thead><tbody><tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>S</mi> <mi>n</mi></msup><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[S^n,-]</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>A</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[-,A]</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⊗</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">(-) \otimes A</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/covariant+functor">covariant</a> <a class="existingWikiWord" href="/nlab/show/hom+functor">hom</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/contravariant+functor">contravariant</a> <a class="existingWikiWord" href="/nlab/show/hom+functor">hom</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Ext">Ext</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Ext">Ext</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Tor">Tor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/end">end</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/end">end</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coend">coend</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/derived+hom+space">derived hom space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mi>Hom</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mi>n</mi></msup><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}Hom(S^n,-)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cocycles">cocycles</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mi>Hom</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}Hom(-,A)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/derived+tensor+product">derived tensor product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msup><mo>⊗</mo> <mi>𝕃</mi></msup><mi>A</mi></mrow><annotation encoding="application/x-tex">(-) \otimes^{\mathbb{L}} A</annotation></semantics></math></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <p>Original discussion:</p> <ul> <li id="EilenbergMacLane42"> <p><a class="existingWikiWord" href="/nlab/show/Samuel+Eilenberg">Samuel Eilenberg</a>, <a class="existingWikiWord" href="/nlab/show/Saunders+MacLane">Saunders MacLane</a>, <em>Group Extensions and Homology</em>, Annals of Mathematics <strong>43</strong> 4 (1942) 757-831 [<a href="https://doi.org/10.2307/1968966">doi:10.2307/1968966</a>, <a href="https://www.jstor.org/stable/1968966">jstor:1968966</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Nobuo+Yoneda">Nobuo Yoneda</a>, <em>On ext and exact sequences</em>, PhD thesis, Journal of the Faculty of Science, Imperial University of Tokyo, 1960 (<a href="http://alpha.math.uga.edu/~lorenz/YonedaExtExactSequences.pdf">pdf</a>, <a href="https://ci.nii.ac.jp/naid/500000325773">CiNii:naid/500000325773</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Saunders+MacLane">Saunders MacLane</a>, <em>Group Extensions by primary abelian groups</em>, Transactions of the American Mathematical Society, <strong>95</strong> 1 (1960) 1-16 [<a href="http://www.jstor.org/stable/1993327">jstor:1993327</a>]</p> </li> </ul> <p>Texbook accounts (see also most references at <em><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a></em>):</p> <ul> <li id="Weibel"> <p><a class="existingWikiWord" href="/nlab/show/Charles+Weibel">Charles Weibel</a>, <em><a class="existingWikiWord" href="/nlab/show/An+Introduction+to+Homological+Algebra">An Introduction to Homological Algebra</a></em>, Cambridge Studies in Adv. Math. 38, CUP 1994</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Henri+Cartan">Henri Cartan</a>, <a class="existingWikiWord" href="/nlab/show/Samuel+Eilenberg">Samuel Eilenberg</a>, <em>Homological algebra</em>, Princeton Univ. Press 1956.</p> </li> <li> <p>S. I . Gelfand, <a class="existingWikiWord" href="/nlab/show/Yuri+Manin">Yuri Manin</a>, <em>Methods of homological algebra</em></p> </li> </ul> <p>A systematic discussion from the point of view of <a class="existingWikiWord" href="/nlab/show/derived+categories">derived categories</a> is in</p> <ul> <li id="KashiwaraShapira"><a class="existingWikiWord" href="/nlab/show/Masaki+Kashiwara">Masaki Kashiwara</a>, <a class="existingWikiWord" href="/nlab/show/Pierre+Schapira">Pierre Schapira</a>, <em><a class="existingWikiWord" href="/nlab/show/Categories+and+Sheaves">Categories and Sheaves</a></em>, Springer (2000)</li> </ul> <p>Lecture notes:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Kiyoshi+Igusa">Kiyoshi Igusa</a>, <em>25 The Ext Functor</em> (<a href="http://people.brandeis.edu/~igusa/Math101b/Ext.pdf">pdf</a>)</p> </li> <li id="Boardman"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Boardman">Michael Boardman</a>, <em>Some Common <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Tor</mi></mrow><annotation encoding="application/x-tex">Tor</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ext</mi></mrow><annotation encoding="application/x-tex">Ext</annotation></semantics></math> Groups</em> (<a href="http://math.jhu.edu/~jmb/note/torext.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/BoardmanTorAndExtGroups.pdf" title="pdf">pdf</a>)</p> </li> <li id="May"> <p><a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, Section 4 of: <em>Notes on Tor and Ext</em> (<a href="http://www.math.uchicago.edu/~may/MISC/TorExt.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Patrick+Morandi">Patrick Morandi</a>, <em>Ext Groups and Ext Functors</em>, (<a href="http://sierra.nmsu.edu/morandi/oldwebpages/math683fall2002/Ext.pdf">pdf</a>)</p> <p>(warning: the last section on resolutions for cocycles for group (abelian) extensions is not correct)</p> </li> </ul> <p>See also:</p> <ul> <li>Wikipedia, <em><a href="http://en.wikipedia.org/wiki/Ext_functor">Ext functor</a></em></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on August 23, 2023 at 14:15:44. 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