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width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/6999/#Item_5" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body><hr /> <p>This entry discusses the general notion of <em><a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a></em> specified to the special context of <a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a>, hence to functors between <a class="existingWikiWord" href="/nlab/show/categories+of+chain+complexes">categories of chain complexes</a>.</p> <p>In the literature this is often understood to be the default case of derived functors. For more discussion of how the following relates to the more general concepts of derived functors see at <em><a href="derived+functor#InHomologicalAlgebra">derived functor – In homological algebra</a></em>.</p> <hr /> <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> <li><a href='#Properties'>Properties</a></li> <ul> <li><a href='#basic_properties'>Basic properties</a></li> <li><a href='#LongExactSequence'>Long exact sequence of a derived functor</a></li> <li><a href='#ViaAcyclicResolutions'>Via acyclic resolutions</a></li> <li><a href='#derived_adjoint_functors'>Derived adjoint functors</a></li> <ul> <li><a href='#observation'>Observation</a></li> </ul> <li><a href='#preservation_of_further_limits_and_colimits'>Preservation of further limits and colimits</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>The general concept of <a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a> is in homological algebra usually called the <em><a class="existingWikiWord" href="/nlab/show/total+derived+functor">total</a> <a class="existingWikiWord" href="/nlab/show/hyper-derived+functor">hyper-derived functor</a></em>, with just “derived functor” being reserved for a more restrictive case. In this tradition, we consider the special case first and then generalize it in stages. The relation between all these notions is discussed <a href="#Properties">below</a>.</p> <h2 id="definition">Definition</h2> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>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>. Without essential loss of generality, we may assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo>=</mo><mi>R</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">\mathcal{A} = R Mod</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/category+of+modules">category of modules</a> over some <a class="existingWikiWord" href="/nlab/show/ring">ring</a>, and we will often speak in terms of this case.</p> <p>The category <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> embeds into its <a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a>, the category of degreewise <a class="existingWikiWord" href="/nlab/show/injective+object">injective</a> <a class="existingWikiWord" href="/nlab/show/cochain+complexes">cochain complexes</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>:</mo><mi>𝒜</mi><mo>→</mo><msup><mi>𝒟</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>𝒦</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msub><mi>ℐ</mi> <mi>𝒜</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> P : \mathcal{A} \to \mathcal{D}^\bullet(\mathcal{A}) = \mathcal{K}^\bullet(\mathcal{I}_{\mathcal{A}}) </annotation></semantics></math></div> <p>or degreewise <a class="existingWikiWord" href="/nlab/show/projective+object">projective</a> <a class="existingWikiWord" href="/nlab/show/chain+complexes">chain complexes</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mo>:</mo><mi>𝒜</mi><mo>→</mo><msub><mi>𝒟</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>𝒦</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>𝒫</mi> <mi>𝒜</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Q : \mathcal{A} \to \mathcal{D}_\bullet(\mathcal{A}) = \mathcal{K}_\bullet(\mathcal{P}_{\mathcal{A}}) </annotation></semantics></math></div> <p>modulo <a class="existingWikiWord" href="/nlab/show/chain+homotopy">chain homotopy</a>. This construction of the derived category naturally gives rise to the following notion of <a class="existingWikiWord" href="/nlab/show/derived+functors">derived functors</a>, which we now discuss.</p> <div class="num_defn" id="AdditiveFunctor"> <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>𝒜</mi><mo>,</mo><mi>ℬ</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}, \mathcal{B}</annotation></semantics></math> two <a class="existingWikiWord" href="/nlab/show/abelian+categories">abelian categories</a> (e.g. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Mod">Mod</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">R'</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Mod">Mod</a>), a <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>𝒜</mi><mo>→</mo><mi>ℬ</mi></mrow><annotation encoding="application/x-tex"> F \colon \mathcal{A} \to \mathcal{B} </annotation></semantics></math></div> <p>is called an <strong><a class="existingWikiWord" href="/nlab/show/additive+functor">additive functor</a></strong> if</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> maps <a class="existingWikiWord" href="/nlab/show/generalized+the">the</a> <a class="existingWikiWord" href="/nlab/show/zero+object">zero object</a> to the zero object, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>≃</mo><mn>0</mn><mo>∈</mo><mi>ℬ</mi></mrow><annotation encoding="application/x-tex">F(0) \simeq 0 \in \mathcal{B}</annotation></semantics></math>;</p> </li> <li> <p>given any two <a class="existingWikiWord" href="/nlab/show/objects">objects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">x, y \in \mathcal{A}</annotation></semantics></math>, there is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo>⊕</mo><mi>y</mi><mo stretchy="false">)</mo><mo>≅</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⊕</mo><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(x \oplus y) \cong F(x) \oplus F(y)</annotation></semantics></math>, and this respects the inclusion and projection maps of the <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a>:</p> </li> </ol> <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>x</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>i</mi> <mi>x</mi></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><msub><mi>i</mi> <mi>y</mi></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>x</mi><mo>⊕</mo><mi>y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>p</mi> <mi>x</mi></msub></mrow></mpadded></msup><mo>↙</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><msub><mi>p</mi> <mi>y</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>x</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>y</mi></mtd></mtr></mtable></mrow><mspace width="1em"></mspace><mspace width="1em"></mspace><mover><mo>↦</mo><mi>F</mi></mover><mspace width="1em"></mspace><mspace width="1em"></mspace><mrow><mtable><mtr><mtd><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>i</mi> <mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><msub><mi>i</mi> <mrow><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo>⊕</mo><mi>y</mi><mo stretchy="false">)</mo><mo>≅</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⊕</mo><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>p</mi> <mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub></mrow></mpadded></msup><mo>↙</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><msub><mi>p</mi> <mrow><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array { x & & & & y \\ & {}_{\mathllap{i_x}}\searrow & & \swarrow_{\mathrlap{i_y}} \\ & & x \oplus y \\ & {}^{\mathllap{p_x}}\swarrow & & \searrow^{\mathrlap{p_y}} \\ x & & & & y } \quad\quad\stackrel{F}{\mapsto}\quad\quad \array { F(x) & & & & F(y) \\ & {}_{\mathllap{i_{F(x)}}}\searrow & & \swarrow_{\mathrlap{i_{F(y)}}} \\ & & F(x \oplus y) \cong F(x) \oplus F(y) \\ & {}^{\mathllap{p_{F(x)}}}\swarrow & & \searrow^{\mathrlap{p_{F(y)}}} \\ F(x) & & & & F(y) } </annotation></semantics></math></div></div> <div class="num_defn" id="ProlongationOfFunctorToChainComplexes"> <h6 id="definition_3">Definition</h6> <p>Given an <a class="existingWikiWord" href="/nlab/show/additive+functor">additive functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>𝒜</mi><mo>→</mo><mi>𝒜</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">F : \mathcal{A} \to \mathcal{A}'</annotation></semantics></math>, it canonically induces a functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</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>𝒜</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Ch_\bullet(F) \colon Ch_\bullet(\mathcal{A}) \to Ch_\bullet(\mathcal{A}') </annotation></semantics></math></div> <p>between <a class="existingWikiWord" href="/nlab/show/categories+of+chain+complexes">categories of chain complexes</a> (its “prolongation”) by applying it to each <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> and to all the diagrams in the definition of a <a class="existingWikiWord" href="/nlab/show/chain+map">chain map</a>. Similarly it preserves <a class="existingWikiWord" href="/nlab/show/chain+homotopies">chain homotopies</a> and hence it passes to the quotient given by the strong <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+chain+complexes">homotopy category of chain complexes</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒦</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>𝒦</mi><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo><mo>→</mo><mi>𝒦</mi><mo stretchy="false">(</mo><mi>𝒜</mi><mo>′</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{K}(F) \colon \mathcal{K}(\mathcal{A}) \to \mathcal{K}(\mathcal{A}') \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>If <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> 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> have <a class="existingWikiWord" href="/nlab/show/projective+object">enough projectives</a>, then their <a class="existingWikiWord" href="/nlab/show/derived+categories">derived categories</a> are</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝒟</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>𝒦</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>𝒫</mi> <mi>𝒜</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathcal{D}_\bullet(\mathcal{A}) \simeq \mathcal{K}_\bullet(\mathcal{P}_{\mathcal{A}}) </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><msup><mi>𝒟</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>𝒦</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msub><mi>ℐ</mi> <mi>𝒜</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathcal{D}^\bullet(\mathcal{A}) \simeq \mathcal{K}^\bullet(\mathcal{I}_{\mathcal{A}}) </annotation></semantics></math></div> <p>etc. One wants to accordingly <em>derive</em> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒟</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>𝒟</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>𝒜</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{D}_\bullet(\mathcal{A}) \to \mathcal{D}_\bullet(\mathcal{A}')</annotation></semantics></math> between these derived categories. It is immediate to achieve this on the domain category, there we can simply precompose and form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo>→</mo><msub><mi>𝒟</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>𝒦</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>𝒫</mi> <mi>𝒜</mi></msub><mo stretchy="false">)</mo><mo>↪</mo><msub><mi>𝒦</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>𝒦</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mrow></mover><msub><mi>𝒦</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>𝒜</mi><mo>′</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{A} \to \mathcal{D}_\bullet(\mathcal{A}) \simeq \mathcal{K}_\bullet(\mathcal{P}_{\mathcal{A}}) \hookrightarrow \mathcal{K}_\bullet(\mathcal{A}) \stackrel{\mathcal{K}_\bullet(F)}{\to} \mathcal{K}_\bullet(\mathcal{A}') \,. </annotation></semantics></math></div> <p>But the resulting composite lands in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒦</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>𝒜</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{K}_\bullet(\mathcal{A}')</annotation></semantics></math> and in general does not factor through the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒟</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>𝒜</mi><mo>′</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>𝒦</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>𝒫</mi> <mrow><mi>𝒜</mi><mo>′</mo></mrow></msub><mo stretchy="false">)</mo><mo>↪</mo><msub><mi>𝒦</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>𝒜</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{D}_\bullet(\mathcal{A}') = \mathcal{K}_\bullet(\mathcal{P}_{\mathcal{A}'}) \hookrightarrow \mathcal{K}_\bullet(\mathcal{A}')</annotation></semantics></math>.</p> <p>In a more general abstract discussion than we present here, one finds that by applying a projective resolution functor <em>on chain complexes</em>, one can enforce this factorization. However, by definition of <a class="existingWikiWord" href="/nlab/show/resolution">resolution</a>, the resulting chain complex is <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphic</a> to the one obtained by the above composite.</p> <p>This means that if one is only interested in the “weak chain homology type” of the chain complex in the image of a <a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a>, then forming <a class="existingWikiWord" href="/nlab/show/chain+homology">chain homology</a> groups of the chain complexes in the images of the above composite gives the desired information. This is what def. <a class="maruku-ref" href="#RightDerivedFunctorOfLeftExactFunctor"></a> and def. <a class="maruku-ref" href="#LeftDerivedFunctorOfRightExactFunctor"></a> below do.</p> </div> <div class="num_defn" id="LeftRightExactFunctor"> <h6 id="definition_4">Definition</h6> <p>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><mo>′</mo></mrow><annotation encoding="application/x-tex">\mathcal{A}, \mathcal{A}'</annotation></semantics></math> be two <a class="existingWikiWord" href="/nlab/show/abelian+categories">abelian categories</a>, for instance <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo>=</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">\mathcal{A} = R </annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Mod">Mod</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo>′</mo><mo>=</mo><mi>R</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\mathcal{A}' = R'</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Mod">Mod</a>. Then a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>𝒜</mi><mo>→</mo><mi>𝒜</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">F \colon \mathcal{A} \to \mathcal{A}'</annotation></semantics></math> which preserves <a class="existingWikiWord" href="/nlab/show/direct+sums">direct sums</a> (and hence in particular the <a class="existingWikiWord" href="/nlab/show/zero+object">zero object</a>) is called</p> <ul> <li> <p>a <strong><a class="existingWikiWord" href="/nlab/show/left+exact+functor">left exact functor</a></strong> if it preserves <a class="existingWikiWord" href="/nlab/show/kernels">kernels</a>;</p> </li> <li> <p>a <strong><a class="existingWikiWord" href="/nlab/show/right+exact+functor">right exact functor</a></strong> if it preserves <a class="existingWikiWord" href="/nlab/show/cokernels">cokernels</a>;</p> </li> <li> <p>an <strong><a class="existingWikiWord" href="/nlab/show/exact+functor">exact functor</a></strong> if it is both left and right exact.</p> </li> </ul> </div> <p>Here to “preserve kernels” means that for every morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>→</mo><mi>f</mi></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \stackrel{f}{\to} Y</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> we have an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> on the left of the following <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>F</mi><mo stretchy="false">(</mo><mi>ker</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>=</mo></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>=</mo></mpadded></msup></mtd></mtr> <mtr><mtd><mi>ker</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ F(ker(f)) &\to& F(X) & \stackrel{F(f)}{\to} & F(Y) \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{=}} && \downarrow^{\mathrlap{=}} \\ ker(F(f)) &\to& F(X) &\stackrel{F(f)}{\to}& F(Y) } \,, </annotation></semantics></math></div> <p>hence that both rows are <a class="existingWikiWord" href="/nlab/show/exact+sequence">exact</a>. And dually for right exact functors.</p> <p>We record the following immediate consequence of this definition (which in the literature is often taken to be the definition).</p> <div class="num_prop" id="LeftRightExactFunctorsCharacterizedByExactSequences"> <h6 id="proposition">Proposition</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is a left exact functor, then for every <a class="existingWikiWord" href="/nlab/show/exact+sequence">exact sequence</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><mn>0</mn><mo>→</mo><mi>A</mi><mo>→</mo><mi>B</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex"> 0 \to A \to B \to C </annotation></semantics></math></div> <p>also</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>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> 0 \to F(A) \to F(B) \to F(C) </annotation></semantics></math></div> <p>is an exact sequence. Dually, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is a right exact functor, then for every <a class="existingWikiWord" href="/nlab/show/exact+sequence">exact sequence</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>B</mi><mo>→</mo><mi>C</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> A \to B \to C \to 0 </annotation></semantics></math></div> <p>also</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> F(A) \to F(B) \to F(C) \to 0 </annotation></semantics></math></div> <p>is an exact sequence.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>A</mi><mo>→</mo><mi>B</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">0 \to A \to B \to C</annotation></semantics></math> is exact then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>↪</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A \hookrightarrow B</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a>. But then the statement that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>B</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">A \to B \to C</annotation></semantics></math> is exact at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> says precisely that <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 <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">B \to C</annotation></semantics></math>. So if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is left exact then by definition also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(A) \to F(B)</annotation></semantics></math> is the kernel of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(B) \to F(C)</annotation></semantics></math> and so is in particular also a monomorphism. Dually for right exact functors.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>Proposition <a class="maruku-ref" href="#LeftRightExactFunctorsCharacterizedByExactSequences"></a> is clearly the motivation for the terminology in def. <a class="maruku-ref" href="#LeftRightExactFunctor"></a>: a functor is left exact if it preserves short exact sequences to the left, and right exact if it preserves them to the right.</p> </div> <p>Now we can state the main two definitions of this section.</p> <div class="num_defn" id="RightDerivedFunctorOfLeftExactFunctor"> <h6 id="definition_5">Definition</h6> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>𝒜</mi><mo>→</mo><mi>𝒜</mi><mo>′</mo></mrow><annotation encoding="application/x-tex"> F : \mathcal{A} \to \mathcal{A}' </annotation></semantics></math></div> <p>be a <a class="existingWikiWord" href="/nlab/show/left+exact+functor">left exact functor</a> between <a class="existingWikiWord" href="/nlab/show/abelian+categories">abelian categories</a> such that <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> has <a class="existingWikiWord" href="/nlab/show/enough+injectives">enough injectives</a>. 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 <a class="existingWikiWord" href="/nlab/show/right+derived+functor">right derived functor</a></strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>R</mi> <mi>n</mi></msup><mi>F</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒜</mi><mover><mo>→</mo><mi>P</mi></mover><msup><mi>𝒦</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msub><mi>ℐ</mi> <mi>𝒜</mi></msub><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msup><mi>𝒦</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mrow></mover><msup><mi>𝒦</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>𝒜</mi><mo>′</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mover><mi>𝒜</mi><mo>′</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> R^n F \;\colon\; \mathcal{A} \stackrel{P}{\to} \mathcal{K}^\bullet(\mathcal{I}_{\mathcal{A}}) \stackrel{\mathcal{K}^\bullet(F)}{\to} \mathcal{K}^\bullet(\mathcal{A}') \stackrel{H^n(-)}{\to} \mathcal{A}' \,, </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>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/injective+resolution">injective resolution</a> functor;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒦</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{K}^\bullet(F)</annotation></semantics></math> is a prolongation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> analogous to def. <a class="maruku-ref" href="#ProlongationOfFunctorToChainComplexes"></a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^n(-)</annotation></semantics></math> is 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>-<a class="existingWikiWord" href="/nlab/show/chain+homology">chain homology</a> functor. Hence</p> </li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>R</mi> <mi>n</mi></msup><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>X</mi> <mo>•</mo></msup><mo stretchy="false">)</mo><mo>≔</mo><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><msup><mo stretchy="false">)</mo> <mo>•</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (R^n F)(X^\bullet) \coloneqq H^n(F(P(X)^\bullet)) \,. </annotation></semantics></math></div></div> <p>Dually:</p> <div class="num_defn" id="LeftDerivedFunctorOfRightExactFunctor"> <h6 id="definition_6">Definition</h6> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>𝒜</mi><mo>→</mo><mi>𝒜</mi><mo>′</mo></mrow><annotation encoding="application/x-tex"> F \colon \mathcal{A} \to \mathcal{A}' </annotation></semantics></math></div> <p>be a <a class="existingWikiWord" href="/nlab/show/right+exact+functor">right exact functor</a> between <a class="existingWikiWord" href="/nlab/show/abelian+categories">abelian categories</a> such that <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> has <a class="existingWikiWord" href="/nlab/show/enough+projectives">enough projectives</a>. 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 <a class="existingWikiWord" href="/nlab/show/left+derived+functor">left derived functor</a></strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mi>n</mi></msub><mi>F</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒜</mi><mover><mo>→</mo><mi>Q</mi></mover><msub><mi>𝒦</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>𝒫</mi> <mi>𝒜</mi></msub><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>𝒦</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mrow></mover><msub><mi>𝒦</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>𝒜</mi><mo>′</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mover><mi>𝒜</mi><mo>′</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> L_n F \;\colon\; \mathcal{A} \stackrel{Q}{\to} \mathcal{K}_\bullet(\mathcal{P}_{\mathcal{A}}) \stackrel{\mathcal{K}_\bullet(F)}{\to} \mathcal{K}_\bullet(\mathcal{A}') \stackrel{H_n(-)}{\to} \mathcal{A}' \,, </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>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/projective+resolution">projective resolution</a> functor;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒦</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mo>=</mo><mi>𝒦</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{K}_\bullet(F)=\mathcal{K}(F)</annotation></semantics></math> is the prolongation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> according to def. <a class="maruku-ref" href="#ProlongationOfFunctorToChainComplexes"></a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H_n(-)</annotation></semantics></math> is 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>-<a class="existingWikiWord" href="/nlab/show/chain+homology">chain homology</a> functor. Hence</p> </li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>L</mi> <mi>n</mi></msub><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>X</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>≔</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (L_n F)(X_\bullet) \coloneqq H_n(F(Q(X)_\bullet)) \,. </annotation></semantics></math></div></div> <h2 id="Properties">Properties</h2> <h3 id="basic_properties">Basic properties</h3> <p>The following proposition says that in degree 0 these derived functors coincide with the original functors.</p> <div class="num_prop" id="BasicPropertiesOfDerivedFunctors"> <h6 id="proposition_2">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>𝒜</mi><mo>→</mo><mi>ℬ</mi></mrow><annotation encoding="application/x-tex">F \colon \mathcal{A} \to \mathcal{B}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/left+exact+functor">left exact functor</a>, def. <a class="maruku-ref" href="#LeftRightExactFunctor"></a> in the presence of <a class="existingWikiWord" href="/nlab/show/enough+injectives">enough injectives</a>. Then for all <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> there is a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>R</mi> <mn>0</mn></msup><mi>F</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>F</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> R^0F(X) \simeq F(X) \,. </annotation></semantics></math></div> <p>Dually, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/right+exact+functor">right exact functor</a> in the presence of <a class="existingWikiWord" href="/nlab/show/enough+projectives">enough projectives</a>, then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>0</mn></msub><mi>F</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>F</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> L_0 F(X) \simeq F(X) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>We discuss the first statement, the second is formally dual.</p> <p>By <a href="/nlab/show/injective+resolution#InjectiveResolutionInComponents">this</a> remark an injective resolution <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>→</mo><mrow><msub><mo>≃</mo> <mi>qi</mi></msub></mrow></mover><msup><mi>X</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">X \stackrel{\simeq_{qi}}{\to} X^\bullet</annotation></semantics></math> is equivalently an <a class="existingWikiWord" href="/nlab/show/exact+sequence">exact sequence</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><mn>0</mn><mo>→</mo><mi>X</mi><mo>↪</mo><msup><mi>X</mi> <mn>0</mn></msup><mo>→</mo><msup><mi>X</mi> <mn>1</mn></msup><mo>→</mo><mi>⋯</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> 0 \to X \hookrightarrow X^0 \to X^1 \to \cdots \,. </annotation></semantics></math></div> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is left exact then it preserves this exact sequence by definition of left exactness, and hence</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>F</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>F</mi><mo stretchy="false">(</mo><msup><mi>X</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><msup><mi>X</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>→</mo><mi>⋯</mi></mrow><annotation encoding="application/x-tex"> 0 \to F(X) \hookrightarrow F(X^0) \to F(X^1) \to \cdots </annotation></semantics></math></div> <p>is an exact sequence. But this means that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>R</mi> <mn>0</mn></msup><mi>F</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>ker</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><msup><mi>X</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><msup><mi>X</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>F</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> R^0 F(X) \coloneqq ker(F(X^0) \to F(X^1)) \simeq F(X) \,. </annotation></semantics></math></div></div> <p>The following immediate consequence of the definition is worth recording:</p> <div class="num_prop" id="LeftDerivedFunctorOnProjectiveVanishes"> <h6 id="proposition_3">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/additive+functor">additive functor</a>.</p> <ul> <li> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/right+exact+functor">right exact</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>∈</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">N \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><msub><mi>L</mi> <mi>n</mi></msub><mi>F</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∀</mo><mi>n</mi><mo>≥</mo><mn>1</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> L_n F(N) = 0 \;\;\;\; \forall n \geq 1 \,. </annotation></semantics></math></div></li> <li> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/left+exact+functor">left exact</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>∈</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">N \in \mathcal{A}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/injective+object">injective 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>R</mi> <mi>n</mi></msup><mi>F</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∀</mo><mi>n</mi><mo>≥</mo><mn>1</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> R^n F(N) = 0 \;\;\;\; \forall n \geq 1 \,. </annotation></semantics></math></div></li> </ul> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>If <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 projective then the chain complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>⋯</mi><mo>→</mo><mn>0</mn><mo>→</mo><mn>0</mn><mo>→</mo><mi>N</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\cdots \to 0 \to 0 \to N]</annotation></semantics></math> is already a <a class="existingWikiWord" href="/nlab/show/projective+resolution">projective resolution</a> and hence by definition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mi>n</mi></msub><mi>F</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L_n F(N) \simeq H_n(0)</annotation></semantics></math> 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 \geq 1</annotation></semantics></math>. Dually if <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 an injective object.</p> </div> <p>For proving the basic property of derived functors below in prop. <a class="maruku-ref" href="#LongExactSequenceOfRightDerivedFunctorsFromShortExactSequence"></a> which continues these basis statements to higher degree, in a certain way, we need the following technical lemma.</p> <div class="num_lemma" id="ProjectiveResolutionOfExactSequenceByExactSequence"> <h6 id="lemma">Lemma</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>A</mi><mover><mo>→</mo><mi>i</mi></mover><mi>B</mi><mover><mo>→</mo><mi>p</mi></mover><mi>C</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">0 \to A \stackrel{i}{\to} B \stackrel{p}{\to} C \to 0</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a> in an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a> with <a class="existingWikiWord" href="/nlab/show/enough+projectives">enough projectives</a>, there exists a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> of <a class="existingWikiWord" href="/nlab/show/chain+complexes">chain complexes</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><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>A</mi> <mo>•</mo></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>B</mi> <mo>•</mo></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>C</mi> <mo>•</mo></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>f</mi> <mo>•</mo></msub></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>g</mi> <mo>•</mo></msub></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>h</mi> <mo>•</mo></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>A</mi></mtd> <mtd><mover><mo>→</mo><mi>i</mi></mover></mtd> <mtd><mi>B</mi></mtd> <mtd><mover><mo>→</mo><mi>p</mi></mover></mtd> <mtd><mi>C</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>0</mn></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ 0 &\to& A_\bullet &\to& B_\bullet &\to& C_\bullet &\to& 0 \\ && \downarrow^{\mathrlap{f_\bullet}} && \downarrow^{\mathrlap{g_\bullet}} && \downarrow^{\mathrlap{h_\bullet}} \\ 0 &\to& A &\stackrel{i}{\to}& B &\stackrel{p}{\to}& C &\to& 0 } </annotation></semantics></math></div> <p>where</p> <ul> <li>each vertical morphism is a <a class="existingWikiWord" href="/nlab/show/projective+resolution">projective resolution</a>;</li> </ul> <p>and in addition</p> <ul> <li>the top row is again a <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a> of chain complexes.</li> </ul> </div> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>By prop. <a class="maruku-ref" href="#ExistenceOfInjectiveResolutions"></a> we can choose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">f_\bullet</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>h</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">h_\bullet</annotation></semantics></math>. The task is now to construct the third resolution <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">g_\bullet</annotation></semantics></math> such as to obtain a short exact sequence of chain complexes, hence degreewise a short exact sequence, in the two row.</p> <p>To construct this, let for each <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></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mi>n</mi></msub><mo>≔</mo><msub><mi>A</mi> <mi>n</mi></msub><mo>⊕</mo><msub><mi>C</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex"> B_n \coloneqq A_n \oplus C_n </annotation></semantics></math></div> <p>be the <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> and let the top horizontal morphisms be the canonical inclusion and projection maps of the direct sum.</p> <p>Let then furthermore (in <a class="existingWikiWord" href="/nlab/show/matrix+calculus">matrix calculus</a> notation)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mn>0</mn></msub><mo>=</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>0</mn></msub><msub><mo stretchy="false">)</mo> <mi>A</mi></msub></mtd> <mtd><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>0</mn></msub><msub><mo stretchy="false">)</mo> <mi>B</mi></msub></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mo>:</mo><msub><mi>A</mi> <mn>0</mn></msub><mo>⊕</mo><msub><mi>C</mi> <mn>0</mn></msub><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex"> g_0 = \left( \array{ (g_0)_A & (g_0)_B } \right) : A_0 \oplus C_0 \to B </annotation></semantics></math></div> <p>be given in the first component by the given composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>0</mn></msub><msub><mo stretchy="false">)</mo> <mi>A</mi></msub><mo>:</mo><msub><mi>A</mi> <mn>0</mn></msub><mo>⊕</mo><msub><mi>C</mi> <mn>0</mn></msub><mover><mo>→</mo><mrow></mrow></mover><msub><mi>A</mi> <mn>0</mn></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mn>0</mn></msub></mrow></mover><mi>A</mi><mover><mo>↪</mo><mi>i</mi></mover><mi>B</mi></mrow><annotation encoding="application/x-tex"> (g_0)_A : A_0 \oplus C_0 \stackrel{}{\to} A_0 \stackrel{f_0}{\to} A \stackrel{i}{\hookrightarrow} B </annotation></semantics></math></div> <p>and in the second component we take</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>0</mn></msub><msub><mo stretchy="false">)</mo> <mi>C</mi></msub><mo>:</mo><msub><mi>A</mi> <mn>0</mn></msub><mo>⊕</mo><msub><mi>C</mi> <mn>0</mn></msub><mo>→</mo><msub><mi>C</mi> <mn>0</mn></msub><mover><mo>→</mo><mi>ζ</mi></mover><mi>B</mi></mrow><annotation encoding="application/x-tex"> (g_0)_C : A_0 \oplus C_0 \to C_0 \stackrel{\zeta}{\to} B </annotation></semantics></math></div> <p>to be given by a lift in</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><mi>B</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>ζ</mi></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>C</mi> <mn>0</mn></msub></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>h</mi> <mn>0</mn></msub></mrow></mover></mtd> <mtd><mi>C</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ && B \\ & {}^{\mathllap{\zeta}}\nearrow & \downarrow^{\mathrlap{p}} \\ C_0 &\stackrel{h_0}{\to}& C } \,, </annotation></semantics></math></div> <p>which exists by the <a class="existingWikiWord" href="/nlab/show/left+lifting+property">left lifting property</a> of the <a class="existingWikiWord" href="/nlab/show/projective+object">projective object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">C_0</annotation></semantics></math> (since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">C_\bullet</annotation></semantics></math> is a projective resolution) against the <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>B</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">p : B \to C</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a>.</p> <p>In total this gives in degree 0</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><msub><mi>A</mi> <mn>0</mn></msub></mtd> <mtd><mo>↪</mo></mtd> <mtd><msub><mi>A</mi> <mn>0</mn></msub><mo>⊕</mo><msub><mi>C</mi> <mn>0</mn></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>C</mi> <mn>0</mn></msub></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>f</mi> <mn>0</mn></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>0</mn></msub><msub><mo stretchy="false">)</mo> <mi>A</mi></msub><mo>,</mo><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>0</mn></msub><msub><mo stretchy="false">)</mo> <mi>C</mi></msub><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>↙</mo> <mi>ζ</mi></msub></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>h</mi> <mn>0</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>↪</mo><mi>i</mi></mover></mtd> <mtd><mi>B</mi></mtd> <mtd><mover><mo>→</mo><mi>p</mi></mover></mtd> <mtd><mi>C</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A_0 &\hookrightarrow& A_0 \oplus C_0 &\to& C_0 \\ {}^{\mathllap{f_0}}\downarrow && {}^{\mathllap{((g_0)_A, (g_0)_C)}}\downarrow &\swarrow_{\zeta}& \downarrow^{\mathrlap{h_0}} \\ A &\stackrel{i}{\hookrightarrow}& B &\stackrel{p}{\to}& C } \,. </annotation></semantics></math></div> <p>Let then the <a class="existingWikiWord" href="/nlab/show/differentials">differentials</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">B_\bullet</annotation></semantics></math> be given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>d</mi> <mi>k</mi> <mrow><msub><mi>B</mi> <mo>•</mo></msub></mrow></msubsup><mo>=</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><msubsup><mi>d</mi> <mi>k</mi> <mrow><msub><mi>A</mi> <mo>•</mo></msub></mrow></msubsup></mtd> <mtd><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>k</mi></msup><msub><mi>e</mi> <mi>k</mi></msub></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><msubsup><mi>d</mi> <mi>k</mi> <mrow><msub><mi>C</mi> <mo>•</mo></msub></mrow></msubsup></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mo>:</mo><msub><mi>A</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>⊕</mo><msub><mi>C</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>→</mo><msub><mi>A</mi> <mi>k</mi></msub><mo>⊕</mo><msub><mi>C</mi> <mi>k</mi></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> d_k^{B_\bullet} = \left( \array{ d_k^{A_\bullet} & (-1)^k e_k \\ 0 & d_k^{C_\bullet} } \right) : A_{k+1} \oplus C_{k+1} \to A_k \oplus C_k \,, </annotation></semantics></math></div> <p>where the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>e</mi> <mi>k</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{e_k\}</annotation></semantics></math> are constructed by <a class="existingWikiWord" href="/nlab/show/induction">induction</a> as follows. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">e_0</annotation></semantics></math> be a lift in</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></mtd> <mtd><msub><mi>A</mi> <mn>0</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>e</mi> <mn>0</mn></msub></mrow></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>f</mi> <mn>0</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>ζ</mi><mo>∘</mo><msubsup><mi>d</mi> <mn>0</mn> <mrow><msub><mi>C</mi> <mo>•</mo></msub></mrow></msubsup><mo lspace="verythinmathspace">:</mo></mtd> <mtd><msub><mi>C</mi> <mn>1</mn></msub></mtd> <mtd><mover><mo>→</mo><mrow></mrow></mover></mtd> <mtd><mi>A</mi></mtd> <mtd><mo>↪</mo><mi>B</mi></mtd> <mtd></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ & && A_0 \\ & & {}^{\mathllap{e_0}}\nearrow & \downarrow^{\mathrlap{f_0}} \\ \zeta \circ d^{C_\bullet}_0 \colon & C_1 &\stackrel{}{\to}& A &\hookrightarrow B& } </annotation></semantics></math></div> <p>which exists since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">C_1</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/projective+object">projective object</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>0</mn></msub><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">A_0 \to A</annotation></semantics></math> is an epimorphism by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">A_\bullet</annotation></semantics></math> being a projective resolution. Here we are using that by exactness the bottom morphism indeed factors through <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 indicated, because the definition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math> and the chain complex property of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">C_\bullet</annotation></semantics></math> gives</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>p</mi><mo>∘</mo><mi>ζ</mi><mo>∘</mo><msubsup><mi>d</mi> <mn>0</mn> <mrow><msub><mi>C</mi> <mo>•</mo></msub></mrow></msubsup></mtd> <mtd><mo>=</mo><msub><mi>h</mi> <mn>0</mn></msub><mo>∘</mo><msubsup><mi>d</mi> <mn>0</mn> <mrow><msub><mi>C</mi> <mo>•</mo></msub></mrow></msubsup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mn>0</mn><mo>∘</mo><msub><mi>h</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} p \circ \zeta \circ d^{C_\bullet}_0 &= h_0 \circ d^{C_\bullet}_0 \\ & = 0 \circ h_1 \\ & = 0 \end{aligned} \,. </annotation></semantics></math></div> <p>Now in the induction step, assuming that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">e_{n-1}</annotation></semantics></math> has been been found satisfying the chain complex property, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">e_n</annotation></semantics></math> be a lift in</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></mtd> <mtd><msub><mi>A</mi> <mi>n</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>e</mi> <mi>n</mi></msub></mrow></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msubsup><mi>d</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow> <mrow><msub><mi>A</mi> <mo>•</mo></msub></mrow></msubsup></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>e</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>∘</mo><msubsup><mi>d</mi> <mi>n</mi> <mrow><msub><mi>C</mi> <mo>•</mo></msub></mrow></msubsup><mo lspace="verythinmathspace">:</mo></mtd> <mtd><msub><mi>C</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd> <mtd><mover><mo>↪</mo><mrow></mrow></mover></mtd> <mtd><mi>ker</mi><mo stretchy="false">(</mo><msubsup><mi>d</mi> <mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow> <mrow><msub><mi>A</mi> <mo>•</mo></msub></mrow></msubsup><mo stretchy="false">)</mo><mo>=</mo><mi>im</mi><mo stretchy="false">(</mo><msubsup><mi>d</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow> <mrow><msub><mi>A</mi> <mo>•</mo></msub></mrow></msubsup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>A</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ & && A_n \\ & & {}^{\mathllap{e_{n}}}\nearrow & \downarrow^{\mathrlap{d^{A_\bullet}_{n-1}}} \\ e_{n-1}\circ d_n^{C_\bullet} \colon & C_{n+1} &\stackrel{}{\hookrightarrow}& ker(d^{A_\bullet}_{n-2}) = im(d^{A_\bullet}_{n-1})) &\to& A_{n-1} } \,, </annotation></semantics></math></div> <p>which again exists since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">C_{n+1}</annotation></semantics></math> is projective. That the bottom morphism factors as indicated is the chain complex property of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">e_{n-1}</annotation></semantics></math> inside <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>d</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow> <mrow><msub><mi>B</mi> <mo>•</mo></msub></mrow></msubsup></mrow><annotation encoding="application/x-tex">d^{B_\bullet}_{n-1}</annotation></semantics></math>.</p> <p>To see that the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>d</mi> <mrow><msub><mi>B</mi> <mo>•</mo></msub></mrow></msup></mrow><annotation encoding="application/x-tex">d^{B_\bullet}</annotation></semantics></math> defines this way indeed squares to 0 notice that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>d</mi> <mi>n</mi> <mrow><msub><mi>B</mi> <mo>•</mo></msub></mrow></msubsup><mo>∘</mo><msubsup><mi>d</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow> <mrow><msub><mi>B</mi> <mo>•</mo></msub></mrow></msubsup><mo>=</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mn>0</mn></mtd> <mtd><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>n</mi></msup><mrow><mo>(</mo><msub><mi>e</mi> <mi>n</mi></msub><mo>∘</mo><msubsup><mi>d</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow> <mrow><msub><mi>C</mi> <mo>•</mo></msub></mrow></msubsup><mo>−</mo><msubsup><mi>d</mi> <mi>n</mi> <mrow><msub><mi>A</mi> <mo>•</mo></msub></mrow></msubsup><mo>∘</mo><msub><mi>e</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo></mrow></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mn>0</mn></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> d^{B_\bullet}_{n} \circ d^{B_\bullet}_{n+1} = \left( \array{ 0 & (-1)^{n}\left(e_{n} \circ d^{C_\bullet}_{n+1} - d^{A_\bullet}_n \circ e_{n+1} \right) \\ 0 & 0 } \right) \,. </annotation></semantics></math></div> <p>This vanishes by the very commutativity of the above diagram.</p> <p>This establishes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">g_\bullet</annotation></semantics></math> such that the above diagram commutes and the bottom row is degreewise a short exact sequence, in fact a <a class="existingWikiWord" href="/nlab/show/split+exact+sequence">split exact sequence</a>, by construction.</p> <p>To see that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">g_\bullet</annotation></semantics></math> is indeed a quasi-isomorphism, consider the <a class="existingWikiWord" href="/nlab/show/homology+long+exact+sequence">homology long exact sequence</a> associated to the short exact sequence of cochain complexes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><msub><mi>A</mi> <mo>•</mo></msub><mo>→</mo><msub><mi>B</mi> <mo>•</mo></msub><mo>→</mo><msub><mi>C</mi> <mo>•</mo></msub><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0</annotation></semantics></math>. In positive degrees it implies that the chain homology of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">B_\bullet</annotation></semantics></math> indeed vanishes. In degree 0 it gives the short sequence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>A</mi><mo>→</mo><msub><mi>H</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><msub><mi>B</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>→</mo><mi>B</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">0 \to A \to H_0(B_\bullet) \to B\to 0</annotation></semantics></math> sitting in a commuting diagram</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><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>A</mi></mtd> <mtd><mo>↪</mo></mtd> <mtd><msub><mi>H</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><msub><mi>B</mi> <mo>•</mo></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>C</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>=</mo></mpadded></msup></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>=</mo></mpadded></msup></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>A</mi></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>B</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>C</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>0</mn><mspace width="thinmathspace"></mspace><mo>,</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ 0 &\to& A &\hookrightarrow& H_0(B_\bullet) &\to& C &\to& 0 \\ \downarrow && \downarrow^{\mathrlap{=}} && \downarrow && \downarrow^{\mathrlap{=}} && \downarrow \\ 0 &\to& A &\hookrightarrow& B &\to& C &\to& 0 \,, } </annotation></semantics></math></div> <p>where both rows are exact. That the middle vertical morphism is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> then follows by the <a class="existingWikiWord" href="/nlab/show/five+lemma">five lemma</a>.</p> </div> <p>The formally dual statement to lemma <a class="maruku-ref" href="#ProjectiveResolutionOfExactSequenceByExactSequence"></a> is the following.</p> <div class="num_lemma" id="InjectiveResolutionOfExactSequenceByExactSequence"> <h6 id="lemma_2">Lemma</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>A</mi><mo>→</mo><mi>B</mi><mo>→</mo><mi>C</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">0 \to A \to B \to C \to 0</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a> in an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a> with <a class="existingWikiWord" href="/nlab/show/injective+object">enough injectives</a>, there exists a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> of cochain complexes</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><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>A</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>B</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>C</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mi>A</mi> <mo>•</mo></msup></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mi>B</mi> <mo>•</mo></msup></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mi>C</mi> <mo>•</mo></msup></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>0</mn></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ 0 &\to& A &\to& B &\to& C &\to& 0 \\ && \downarrow^{\mathrlap{}} && \downarrow^{\mathrlap{}} && \downarrow^{\mathrlap{}} \\ 0 &\to& A^\bullet &\to& B^\bullet &\to& C^\bullet &\to& 0 } </annotation></semantics></math></div> <p>where</p> <ul> <li>each vertical morphism is an <a class="existingWikiWord" href="/nlab/show/injective+resolution">injective resolution</a>;</li> </ul> <p>and in addition</p> <ul> <li>the bottom row is again a <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a> of cochain complexes.</li> </ul> </div> <h3 id="LongExactSequence">Long exact sequence of a derived functor</h3> <p>The central general fact about derived functors to be discussed here is now the following.</p> <div class="num_prop" id="LongExactSequenceOfRightDerivedFunctorsFromShortExactSequence"> <h6 id="proposition_4">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo>,</mo><mi>ℬ</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}, \mathcal{B}</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/abelian+categories">abelian categories</a> and assume that <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> has <a class="existingWikiWord" href="/nlab/show/injective+object">enough injectives</a>.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>𝒜</mi><mo>→</mo><mi>ℬ</mi></mrow><annotation encoding="application/x-tex">F : \mathcal{A} \to \mathcal{B}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/left+exact+functor">left exact functor</a> and let</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>B</mi><mo>→</mo><mi>C</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0 \to A \to B \to C \to 0 </annotation></semantics></math></div> <p>be a <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</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>.</p> <p>Then there is a <a class="existingWikiWord" href="/nlab/show/long+exact+sequence">long exact sequence</a> of images of these objects under the right derived functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>R</mi> <mo>•</mo></msup><mi>F</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R^\bullet F(-)</annotation></semantics></math> of def. <a class="maruku-ref" href="#RightDerivedFunctorOfLeftExactFunctor"></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><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mi>R</mi> <mn>0</mn></msup><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mi>R</mi> <mn>0</mn></msup><mi>F</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mi>R</mi> <mn>0</mn></msup><mi>F</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>δ</mi> <mn>0</mn></msub></mrow></mover></mtd> <mtd><msup><mi>R</mi> <mn>1</mn></msup><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mi>R</mi> <mn>1</mn></msup><mi>F</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mi>R</mi> <mn>1</mn></msup><mi>F</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>δ</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><msup><mi>R</mi> <mn>2</mn></msup><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>⋯</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ 0 &\to& R^0F (A) &\to& R^0 F(B) &\to& R^0 F(C) &\stackrel{\delta_0}{\to}& R^1 F(A) &\to& R^1 F(B) &\to& R^1F(C) &\stackrel{\delta_1}{\to}& R^2 F(A) &\to& \cdots \\ && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ 0 &\to& F(A) &\to& F(B) &\to& F(C) } </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℬ</mi></mrow><annotation encoding="application/x-tex">\mathcal{B}</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>By lemma <a class="maruku-ref" href="#InjectiveResolutionOfExactSequenceByExactSequence"></a> we can find an injective resolution</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><msup><mi>A</mi> <mo>•</mo></msup><mo>→</mo><msup><mi>B</mi> <mo>•</mo></msup><mo>→</mo><msup><mi>C</mi> <mo>•</mo></msup><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0 \to A^\bullet \to B^\bullet \to C^\bullet \to 0 </annotation></semantics></math></div> <p>of the given exact sequence which is itself again an exact sequence of cochain complexes.</p> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">A^n</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/injective+object">injective object</a> for all <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>, its component sequences <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><msup><mi>A</mi> <mi>n</mi></msup><mo>→</mo><msup><mi>B</mi> <mi>n</mi></msup><mo>→</mo><msup><mi>C</mi> <mi>n</mi></msup><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">0 \to A^n \to B^n \to C^n \to 0</annotation></semantics></math> are indeed <a class="existingWikiWord" href="/nlab/show/split+exact+sequences">split exact sequences</a> (see the discussion there). Splitness is preserved by any functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> (and also since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/additive+functor">additive</a> it even preserves the <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> structure that is chosen in the proof of lemma <a class="maruku-ref" href="#ProjectiveResolutionOfExactSequenceByExactSequence"></a>) and so it follows that</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>F</mi><mo stretchy="false">(</mo><msup><mover><mi>A</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msup><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><msup><mover><mi>B</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msup><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><msup><mover><mi>C</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msup><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0 \to F(\tilde A^\bullet) \to F(\tilde B^\bullet) \to F(\tilde C^\bullet) \to 0 </annotation></semantics></math></div> <p>is a again short exact sequence of cochain complexes, now in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℬ</mi></mrow><annotation encoding="application/x-tex">\mathcal{B}</annotation></semantics></math>. Hence we have the corresponding <a class="existingWikiWord" href="/nlab/show/homology+long+exact+sequence">homology long exact sequence</a> from prop. <a class="maruku-ref" href="#HomologyLongExactSequence"></a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>→</mo><msup><mi>H</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><msup><mi>A</mi> <mo>•</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><msup><mi>B</mi> <mo>•</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mi>δ</mi></mover><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><msup><mi>A</mi> <mo>•</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><msup><mi>B</mi> <mo>•</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mi>δ</mi></mover><msup><mi>H</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><msup><mi>A</mi> <mo>•</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><msup><mi>B</mi> <mo>•</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mi>⋯</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \cdots \to H^{n-1}(F(A^\bullet)) \to H^{n-1}(F(B^\bullet)) \to H^{n-1}(F(C^\bullet)) \stackrel{\delta}{\to} H^n(F(A^\bullet)) \to H^n(F(B^\bullet)) \to H^n(F(C^\bullet)) \stackrel{\delta}{\to} H^{n+1}(F(A^\bullet)) \to H^{n+1}(F(B^\bullet)) \to H^{n+1}(F(C^\bullet)) \to \cdots \,. </annotation></semantics></math></div> <p>By construction of the resolutions and by def. <a class="maruku-ref" href="#RightDerivedFunctorOfLeftExactFunctor"></a>, this is equal to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>→</mo><msup><mi>R</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>R</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>F</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>R</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>F</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mi>δ</mi></mover><msup><mi>R</mi> <mi>n</mi></msup><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>R</mi> <mi>n</mi></msup><mi>F</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>R</mi> <mi>n</mi></msup><mi>F</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mi>δ</mi></mover><msup><mi>R</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>R</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>F</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>R</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>F</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><mi>⋯</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \cdots \to R^{n-1}F(A) \to R^{n-1}F(B) \to R^{n-1}F(C) \stackrel{\delta}{\to} R^{n}F(A) \to R^{n}F(B) \to R^{n}F(C) \stackrel{\delta}{\to} R^{n+1}F(A) \to R^{n+1}F(B) \to R^{n+1}F(C) \to \cdots \,. </annotation></semantics></math></div> <p>Finally the equivalence of the first three terms with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(A) \to F(B) \to F(C)</annotation></semantics></math> is given by prop. <a class="maruku-ref" href="#BasicPropertiesOfDerivedFunctors"></a>.</p> </div> <div class="num_remark" id="DerivedFunctorAsObstructionToExactness"> <h6 id="remark_3">Remark</h6> <p>Prop. <a class="maruku-ref" href="#LongExactSequenceOfRightDerivedFunctorsFromShortExactSequence"></a> implies that one way to interpret <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>R</mi> <mn>1</mn></msup><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R^1 F(A)</annotation></semantics></math> is as a “measure for how a <a class="existingWikiWord" href="/nlab/show/left+exact+functor">left exact functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> fails to be an <a class="existingWikiWord" href="/nlab/show/exact+functor">exact functor</a>”. For, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>B</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">A \to B \to C</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a>, this proposition gives the exact sequence</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>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>R</mi> <mn>1</mn></msup><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> 0 \to F(A) \to F(B) \to F(C) \to R^1 F(A) </annotation></semantics></math></div> <p>and hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">0 \to F(A) \to F(B) \to F(C) \to 0</annotation></semantics></math> is a short exact sequence itself precisely if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>R</mi> <mn>1</mn></msup><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≃</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">R^1 F(A) \simeq 0</annotation></semantics></math>.</p> <p>Dually, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/right+exact+functor">right exact functor</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>1</mn></msub><mi>F</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L_1 F (C)</annotation></semantics></math> “measures how <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> fails to be exact” for then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>1</mn></msub><mi>F</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> L_1F (C) \to F(A) \to F(B) \to F(C) \to 0 </annotation></semantics></math></div> <p>is an exact sequence and hence is a short exact sequence precisely if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>1</mn></msub><mi>F</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>≃</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">L_1F(C) \simeq 0</annotation></semantics></math>.</p> </div> <p>Notice that in fact we even have the following statement (following directly from the definition).</p> <div class="num_prop" id="DerivedFunctorOfExactFunctor"> <h6 id="proposition_5">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/additive+functor">additive functor</a> which is an <a class="existingWikiWord" href="/nlab/show/exact+functor">exact functor</a>. Then its left and right derived functors vanish in positive degree:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>R</mi> <mrow><mo>≥</mo><mn>1</mn></mrow></msup><mi>F</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> R^{\geq 1} F = 0 </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mrow><mo>≥</mo><mn>1</mn></mrow></msub><mi>F</mi><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> L_{\geq 1} F = 0 \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_6">Proof</h6> <p>Because an <a class="existingWikiWord" href="/nlab/show/exact+functor">exact functor</a> preserves all <a class="existingWikiWord" href="/nlab/show/exact+sequences">exact sequences</a>. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Y</mi> <mo>•</mo></msub><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">Y_\bullet \to A</annotation></semantics></math> is a projective resolution then also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>Y</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">F(Y)_\bullet</annotation></semantics></math> is exact in all positive degrees, and hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></msub><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mi>H</mi> <mrow><mi>n</mi><mo>≥</mo></mrow></msub><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">L_{n\geq 1} F(A) ) H_{n \geq}(F(Y)) = 0</annotation></semantics></math>. Dually for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>R</mi> <mi>n</mi></msup><mi>F</mi></mrow><annotation encoding="application/x-tex">R^n F</annotation></semantics></math>.</p> </div> <p>Conversely:</p> <div class="num_defn" id="AcyclicObject"> <h6 id="definition_7">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>𝒜</mi><mo>→</mo><mi>ℬ</mi></mrow><annotation encoding="application/x-tex">F \colon \mathcal{A} \to \mathcal{B}</annotation></semantics></math> be a left or right exact additive functor. An object <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> is called an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-<strong><a class="existingWikiWord" href="/nlab/show/acyclic+object">acyclic object</a></strong> if all positive-degree right/left derived functors of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> are zero 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>.</p> </div> <h3 id="ViaAcyclicResolutions">Via acyclic resolutions</h3> <p>We now discuss how the derived functor of an additive functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> may also be computed not necessarily with genuine injective/projective resolutions as in def. <a class="maruku-ref" href="#RightDerivedFunctorOfLeftExactFunctor"></a>, but with (just) “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-injective”/“<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-projective resolutions”.</p> <p>While projective resolutions 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> are <em>sufficient</em> for computing <em>every</em> <a class="existingWikiWord" href="/nlab/show/left+derived+functor">left derived functor</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch_\bullet(\mathcal{A})</annotation></semantics></math> and injective resolutions are sufficient for computing <em>every</em> <a class="existingWikiWord" href="/nlab/show/right+derived+functor">right derived functor</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ch</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch^\bullet(\mathcal{A})</annotation></semantics></math>, if one is interested just in a single functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> then such resolutions may be more than <em>necessary</em>. A weaker kind of resolution which is still sufficient is then often more convenient for applications. These <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-projective resolutions</em> and <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-injective resolutions</em>, respectively, we discuss now. A special case of both are <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/acyclic+resolutions">acyclic resolutions</a></em>.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo>,</mo><mi>ℬ</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}, \mathcal{B}</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/abelian+categories">abelian categories</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>𝒜</mi><mo>→</mo><mi>ℬ</mi></mrow><annotation encoding="application/x-tex">F \colon \mathcal{A} \to \mathcal{B}</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/additive+functor">additive functor</a>.</p> <div class="num_defn" id="FInjectives"> <h6 id="definition_8">Definition</h6> <p>Assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/left+exact+functor">left exact</a>. An <a class="existingWikiWord" href="/nlab/show/additive+category">additive</a> <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℐ</mi><mo>⊂</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{I} \subset \mathcal{A}</annotation></semantics></math> is called <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-injective</strong> (or: consisting of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-injective objects) if</p> <ol> <li> <p>for every object <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> there is a <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mover><mi>A</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">A \to \tilde A</annotation></semantics></math> into an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>A</mi><mo stretchy="false">˜</mo></mover><mo>∈</mo><mi>ℐ</mi><mo>⊂</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\tilde A \in \mathcal{I} \subset \mathcal{A}</annotation></semantics></math>;</p> </li> <li> <p>for every <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>A</mi><mo>→</mo><mi>B</mi><mo>→</mo><mi>C</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">0 \to A \to B \to C \to 0</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> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>∈</mo><mi>ℐ</mi><mo>⊂</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">A, B \in \mathcal{I} \subset \mathcal{A}</annotation></semantics></math> also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>∈</mo><mi>ℐ</mi><mo>⊂</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">C \in \mathcal{I} \subset \mathcal{A}</annotation></semantics></math>;</p> </li> <li> <p>for every <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>A</mi><mo>→</mo><mi>B</mi><mo>→</mo><mi>C</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">0 \to A \to B \to C \to 0</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> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>ℐ</mi><mo>⊂</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">A\in \mathcal{I} \subset \mathcal{A}</annotation></semantics></math> also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">0 \to F(A) \to F(B) \to F(C) \to 0</annotation></semantics></math> is a short exact sequence in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℬ</mi></mrow><annotation encoding="application/x-tex">\mathcal{B}</annotation></semantics></math>.</p> </li> </ol> </div> <p>And dually:</p> <div class="num_defn" id="FProjectives"> <h6 id="definition_9">Definition</h6> <p>Assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/right+exact+functor">right exact</a>. An <a class="existingWikiWord" href="/nlab/show/additive+category">additive</a> <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒫</mi><mo>⊂</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{P} \subset \mathcal{A}</annotation></semantics></math> is called <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-projective</strong> (or: consisting of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-projective objects) if</p> <ol> <li> <p>for every object <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> there is an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>A</mi><mo stretchy="false">˜</mo></mover><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\tilde A \to A</annotation></semantics></math> from an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>A</mi><mo stretchy="false">˜</mo></mover><mo>∈</mo><mi>𝒫</mi><mo>⊂</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\tilde A \in \mathcal{P} \subset \mathcal{A}</annotation></semantics></math>;</p> </li> <li> <p>for every <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>A</mi><mo>→</mo><mi>B</mi><mo>→</mo><mi>C</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">0 \to A \to B \to C \to 0</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> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>,</mo><mi>C</mi><mo>∈</mo><mi>𝒫</mi><mo>⊂</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">B, C \in \mathcal{P} \subset \mathcal{A}</annotation></semantics></math> also <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>𝒜</mi></mrow><annotation encoding="application/x-tex">A \in \mathcal{P} \subset \mathcal{A}</annotation></semantics></math>;</p> </li> <li> <p>for every <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>A</mi><mo>→</mo><mi>B</mi><mo>→</mo><mi>C</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">0 \to A \to B \to C \to 0</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> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>∈</mo><mi>𝒫</mi><mo>⊂</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">C\in \mathcal{P} \subset \mathcal{A}</annotation></semantics></math> also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">0 \to F(A) \to F(B) \to F(C) \to 0</annotation></semantics></math> is a short exact sequence in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℬ</mi></mrow><annotation encoding="application/x-tex">\mathcal{B}</annotation></semantics></math>.</p> </li> </ol> </div> <p>With the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℐ</mi><mo>,</mo><mi>𝒫</mi><mo>⊂</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{I},\mathcal{P}\subset \mathcal{A}</annotation></semantics></math> as above, we say:</p> <div class="num_defn" id="FProjectivesResolution"> <h6 id="definition_10">Definition</h6> <p>For <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>,</p> <ul> <li> <p>an <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-injective resolution</strong> 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> is a <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><msup><mi>I</mi> <mo>•</mo></msup><mo>∈</mo><msup><mi>Ch</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>ℐ</mi><mo stretchy="false">)</mo><mo>⊂</mo><msup><mi>Ch</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">I^\bullet \in Ch^\bullet(\mathcal{I}) \subset Ch^\bullet(\mathcal{A})</annotation></semantics></math> and a <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>→</mo><mrow><msub><mo>≃</mo> <mi>qi</mi></msub></mrow></mover><msup><mi>I</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex"> A \stackrel{\simeq_{qi}}{\to} I^\bullet </annotation></semantics></math></div></li> <li> <p>an <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-projective resolution</strong> 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> is a <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Q</mi> <mo>•</mo></msub><mo>∈</mo><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>𝒫</mi><mo stretchy="false">)</mo><mo>⊂</mo><msup><mi>Ch</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Q_\bullet \in Ch_\bullet(\mathcal{P}) \subset Ch^\bullet(\mathcal{A})</annotation></semantics></math> and a <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Q</mi> <mo>•</mo></msub><mover><mo>→</mo><mrow><msub><mo>≃</mo> <mi>qi</mi></msub></mrow></mover><mi>A</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Q_\bullet \stackrel{\simeq_{qi}}{\to} A \,. </annotation></semantics></math></div></li> </ul> </div> <p>Let now <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> have <a class="existingWikiWord" href="/nlab/show/enough+projectives">enough projectives</a> / <a class="existingWikiWord" href="/nlab/show/enough+injectives">enough injectives</a>, respectively.</p> <div class="num_example" id="FAcyclicObjectsAreFProjectiveObjects"> <h6 id="example">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>𝒜</mi><mo>→</mo><mi>ℬ</mi></mrow><annotation encoding="application/x-tex">F \colon \mathcal{A} \to \mathcal{B}</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/additive+functor">additive functor</a>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ac</mi><mo>⊂</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">Ac \subset \mathcal{A}</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/acyclic+objects">acyclic objects</a>, def. <a class="maruku-ref" href="#AcyclicObject"></a>. Then</p> <ul> <li> <p>if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/left+exact+functor">left exact</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℐ</mi><mo>≔</mo><mi>Ac</mi></mrow><annotation encoding="application/x-tex">\mathcal{I} \coloneqq Ac</annotation></semantics></math> is a subcategory of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-injective objects;</p> </li> <li> <p>if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/right+exact+functor">right exact</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒫</mi><mo>≔</mo><mi>Ac</mi></mrow><annotation encoding="application/x-tex">\mathcal{P} \coloneqq Ac</annotation></semantics></math> is a subcategory of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-projective objects.</p> </li> </ul> </div> <div class="proof"> <h6 id="proof_7">Proof</h6> <p>Consider the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is left exact. The other case works dually. Then the first condition of def. <a class="maruku-ref" href="#FInjectives"></a> is satisfied because every <a class="existingWikiWord" href="/nlab/show/injective+object">injective object</a> is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/acyclic+object">acyclic object</a> and by assumption there are enough of these.</p> <p>For the second and third condition of def. <a class="maruku-ref" href="#FInjectives"></a> use that there is the <a class="existingWikiWord" href="/nlab/show/long+exact+sequence">long exact sequence</a> of <a class="existingWikiWord" href="/nlab/show/derived+functors">derived functors</a> prop. <a class="maruku-ref" href="#LongExactSequenceOfRightDerivedFunctorsFromShortExactSequence"></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>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>R</mi> <mn>1</mn></msup><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>R</mi> <mn>1</mn></msup><mi>F</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>R</mi> <mn>1</mn></msup><mi>F</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>R</mi> <mn>2</mn></msup><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>R</mi> <mn>2</mn></msup><mi>F</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>R</mi> <mn>2</mn></msup><mi>F</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><mo>⋅</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> 0 \to F(A) \to F(B) \to F(C) \to R^1 F(A) \to R^1 F(B) \to R^1 F(C) \to R^2 F(A) \to R^2 F(B) \to R^2 F(C) \to \cdot \,. </annotation></semantics></math></div> <p>For the second condition, by assumption 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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> and definition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/acyclic+object">acyclic object</a> we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>R</mi> <mi>n</mi></msup><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≃</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">R^n F(A) \simeq 0</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>R</mi> <mi>n</mi></msup><mi>F</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>≃</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">R^n F(B) \simeq 0</annotation></semantics></math> 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 \geq 1</annotation></semantics></math> and hence short exact sequences</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><mn>0</mn><mo>→</mo><msup><mi>R</mi> <mi>n</mi></msup><mi>F</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0 \to 0 \to R^n F(C) \to 0 </annotation></semantics></math></div> <p>which imply that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>R</mi> <mi>n</mi></msup><mi>F</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>≃</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">R^n F(C)\simeq 0</annotation></semantics></math> 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>, hence that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is acyclic.</p> <p>Similarly, the third condition is equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>R</mi> <mn>1</mn></msup><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≃</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">R^1 F(A) \simeq 0</annotation></semantics></math>.</p> </div> <div class="num_example" id="FAcyclicResolution"> <h6 id="example_2">Example</h6> <p>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-projective/injective resolutions (def. <a class="maruku-ref" href="#FProjectivesResolution"></a>) by <a class="existingWikiWord" href="/nlab/show/acyclic+objects">acyclic objects</a> as in example <a class="maruku-ref" href="#FAcyclicObjectsAreFProjectiveObjects"></a> are called <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-acyclic resolutions</strong>.</p> </div> <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/injective+object">enough injectives</a>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>𝒜</mi><mo>→</mo><mi>ℬ</mi></mrow><annotation encoding="application/x-tex">F \colon \mathcal{A} \to \mathcal{B}</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/additive+functor">additive</a> <a class="existingWikiWord" href="/nlab/show/left+exact+functor">left exact functor</a> with <a class="existingWikiWord" href="/nlab/show/right+derived+functor">right derived functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>R</mi> <mo>•</mo></msub><mi>F</mi></mrow><annotation encoding="application/x-tex">R_\bullet F</annotation></semantics></math>, def. <a class="maruku-ref" href="#RightDerivedFunctorOfLeftExactFunctor"></a>. Finally let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℐ</mi><mo>⊂</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{I} \subset \mathcal{A}</annotation></semantics></math> be a subcategory of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-injective objects, def. <a class="maruku-ref" href="#FInjectives"></a>.</p> <div class="num_lemma" id="FPreservesNullnessOfFInjectiveComplexes"> <h6 id="lemma_3">Lemma</h6> <p>If a <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><msup><mi>X</mi> <mo>•</mo></msup><mo>∈</mo><msup><mi>Ch</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>ℐ</mi><mo stretchy="false">)</mo><mo>⊂</mo><msup><mi>Ch</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X^\bullet \in Ch^\bullet(\mathcal{I}) \subset Ch^\bullet(\mathcal{A})</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphic</a> to 0,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mo>•</mo></msup><mover><mo>→</mo><mrow><msub><mo>≃</mo> <mi>qi</mi></msub></mrow></mover><mn>0</mn></mrow><annotation encoding="application/x-tex"> X^\bullet \stackrel{\simeq_{qi}}{\to} 0 </annotation></semantics></math></div> <p>then also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><msup><mi>X</mi> <mo>•</mo></msup><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>Ch</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>ℬ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(X^\bullet) \in Ch^\bullet(\mathcal{B})</annotation></semantics></math> is quasi-isomorphic to 0</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><msup><mi>X</mi> <mo>•</mo></msup><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mo>≃</mo> <mi>qi</mi></msub></mrow></mover><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> F(X^\bullet) \stackrel{\simeq_{qi}}{\to} 0 \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_8">Proof</h6> <p>Consider the following collection of <a class="existingWikiWord" href="/nlab/show/short+exact+sequences">short exact sequences</a> obtained from the <a class="existingWikiWord" href="/nlab/show/long+exact+sequence">long exact sequence</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">X^\bullet</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><msup><mi>X</mi> <mn>0</mn></msup><mover><mo>→</mo><mrow><msup><mi>d</mi> <mn>0</mn></msup></mrow></mover><msup><mi>X</mi> <mn>1</mn></msup><mover><mo>→</mo><mrow><msup><mi>d</mi> <mn>1</mn></msup></mrow></mover><mi>im</mi><mo stretchy="false">(</mo><msup><mi>d</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0 \to X^0 \stackrel{d^0}{\to} X^1 \stackrel{d^1}{\to} im(d^1) \to 0 </annotation></semantics></math></div><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>im</mi><mo stretchy="false">(</mo><msup><mi>d</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mi>X</mi> <mn>2</mn></msup><mover><mo>→</mo><mrow><msup><mi>d</mi> <mn>2</mn></msup></mrow></mover><mi>im</mi><mo stretchy="false">(</mo><msup><mi>d</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0 \to im(d^1) \to X^2 \stackrel{d^2}{\to} im(d^2) \to 0 </annotation></semantics></math></div><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>im</mi><mo stretchy="false">(</mo><msup><mi>d</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mi>X</mi> <mn>3</mn></msup><mover><mo>→</mo><mrow><msup><mi>d</mi> <mn>3</mn></msup></mrow></mover><mi>im</mi><mo stretchy="false">(</mo><msup><mi>d</mi> <mn>3</mn></msup><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0 \to im(d^2) \to X^3 \stackrel{d^3}{\to} im(d^3) \to 0 </annotation></semantics></math></div> <p>and so on. Going by <a class="existingWikiWord" href="/nlab/show/induction">induction</a> through this list and using the second condition in def. <a class="maruku-ref" href="#FInjectives"></a> we have that all the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>im</mi><mo stretchy="false">(</mo><msup><mi>d</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">im(d^n)</annotation></semantics></math> are in <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">\mathcal{I}</annotation></semantics></math>. Then the third condition in def. <a class="maruku-ref" href="#FInjectives"></a> says that all the sequences</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>F</mi><mo stretchy="false">(</mo><mi>im</mi><mo stretchy="false">(</mo><msup><mi>d</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><msup><mi>X</mi> <mi>n</mi></msup><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>im</mi><mo stretchy="false">(</mo><msup><mi>d</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0 \to F(im(d^n)) \to F(X^n+1) \to F(im(d^{n+1})) \to 0 </annotation></semantics></math></div> <p>are <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">exact</a>. But this means that</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>F</mi><mo stretchy="false">(</mo><msup><mi>X</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><msup><mi>X</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><msup><mi>X</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo>→</mo><mi>⋯</mi></mrow><annotation encoding="application/x-tex"> 0 \to F(X^0)\to F(X^1) \to F(X^2) \to \cdots </annotation></semantics></math></div> <p>is exact, hence that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><msup><mi>X</mi> <mo>•</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(X^\bullet)</annotation></semantics></math> is quasi-isomorphic to 0.</p> </div> <div class="num_theorem" id="DerivedFFromFInjectiveResolution"> <h6 id="theorem">Theorem</h6> <p>For <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> an object with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-injective resolution <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>→</mo><mrow><msub><mo>≃</mo> <mi>qi</mi></msub></mrow></mover><msubsup><mi>I</mi> <mi>F</mi> <mo>•</mo></msubsup></mrow><annotation encoding="application/x-tex">A \stackrel{\simeq_{qi}}{\to} I_F^\bullet</annotation></semantics></math>, def. <a class="maruku-ref" href="#FProjectivesResolution"></a>, we have for each <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> an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>R</mi> <mi>n</mi></msup><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><msubsup><mi>I</mi> <mi>F</mi> <mo>•</mo></msubsup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> R^n F(A) \simeq H^n(F(I_F^\bullet)) </annotation></semantics></math></div> <p>between 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 right derived functor, def. <a class="maruku-ref" href="#RightDerivedFunctorOfLeftExactFunctor"></a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> evaluated 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> and the <a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> applied to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-injective resolution <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>I</mi> <mi>F</mi> <mo>•</mo></msubsup></mrow><annotation encoding="application/x-tex">I_F^\bullet</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof_9">Proof</h6> <p>By <a href="https://ncatlab.org/schreiber/show/Introduction+to+Homological+algebra#ExistenceOfInjectiveResolutions">this prop.</a> we may also find an injective resolution <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>→</mo><mrow><msub><mo>≃</mo> <mi>qi</mi></msub></mrow></mover><msup><mi>I</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">A \stackrel{\simeq_{qi}}{\to} I^\bullet</annotation></semantics></math>. By <a href="https://ncatlab.org/schreiber/show/Introduction+to+Homological+algebra#InjectiveResolutionOfCodomainRespectsMorphisms">this prop</a> there is a lift of the identity 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> to a <a class="existingWikiWord" href="/nlab/show/chain+map">chain map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>I</mi> <mi>F</mi> <mo>•</mo></msubsup><mo>→</mo><msup><mi>I</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">I^\bullet_F \to I^\bullet</annotation></semantics></math> such that the <a class="existingWikiWord" href="/nlab/show/diagram">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>A</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mo>≃</mo> <mi>qi</mi></msub></mrow></mover></mtd> <mtd><msubsup><mi>I</mi> <mi>F</mi> <mo>•</mo></msubsup></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>id</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mo>≃</mo> <mi>qi</mi></msub></mrow></mover></mtd> <mtd><msup><mi>I</mi> <mo>•</mo></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\stackrel{\simeq_{qi}}{\to}& I_F^\bullet \\ \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{f}} \\ A &\stackrel{\simeq_{qi}}{\to}& I^\bullet } </annotation></semantics></math></div> <p><a class="existingWikiWord" href="/nlab/show/commuting+diagram">commutes</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ch</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch^\bullet(\mathcal{A})</annotation></semantics></math>. Therefore by the <a class="existingWikiWord" href="/nlab/show/2-out-of-3">2-out-of-3</a> property of <a class="existingWikiWord" href="/nlab/show/quasi-isomorphisms">quasi-isomorphisms</a> it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is a quasi-isomorphism</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cone</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>Ch</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cone(f) \in Ch^\bullet(\mathcal{A})</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>I</mi> <mo>•</mo></msup><mo>→</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">I^\bullet \to Cone(f)</annotation></semantics></math> be the canonical <a class="existingWikiWord" href="/nlab/show/chain+map">chain map</a> into it. By the explicit formulas for mapping cones, we have that</p> <ol> <li> <p>there is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(Cone(f)) \simeq Cone(F(f))</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cone</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>Ch</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>ℐ</mi><mo stretchy="false">)</mo><mo>⊂</mo><msup><mi>Ch</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cone(f) \in Ch^\bullet(\mathcal{I})\subset Ch^\bullet(\mathcal{A})</annotation></semantics></math> (because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-injective objects are closed under <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a>).</p> </li> </ol> <p>The first implies that we have a <a class="existingWikiWord" href="/nlab/show/homology+exact+sequence">homology exact sequence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>→</mo><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><msup><mi>I</mi> <mo>•</mo></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><msubsup><mi>I</mi> <mi>F</mi> <mo>•</mo></msubsup><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>f</mi><msup><mo stretchy="false">)</mo> <mo>•</mo></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msup><mi>I</mi> <mo>•</mo></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msubsup><mi>I</mi> <mi>F</mi> <mo>•</mo></msubsup><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>f</mi><msup><mo stretchy="false">)</mo> <mo>•</mo></msup><mo stretchy="false">)</mo><mo>→</mo><mi>⋯</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \cdots \to H^n(I^\bullet) \to H^n(I_F^\bullet) \to H^n(Cone(f)^\bullet) \to H^{n+1}(I^\bullet) \to H^{n+1}(I_F^\bullet) \to H^{n+1}(Cone(f)^\bullet) \to \cdots \,. </annotation></semantics></math></div> <p>Observe that with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">f^\bullet</annotation></semantics></math> a quasi-isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cone</mi><mo stretchy="false">(</mo><msup><mi>f</mi> <mo>•</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cone(f^\bullet)</annotation></semantics></math> is quasi-isomorphic to 0. Therefore the second item above implies with lemma <a class="maruku-ref" href="#FPreservesNullnessOfFInjectiveComplexes"></a> that also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(Cone(f))</annotation></semantics></math> is quasi-isomorphic to 0. This finally means that the above homology exact sequences consists of exact pieces of the form</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><mo stretchy="false">(</mo><msup><mi>R</mi> <mi>n</mi></msup><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≔</mo><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><msup><mi>I</mi> <mo>•</mo></msup><mo stretchy="false">)</mo><mover><mo>→</mo><mo>≃</mo></mover><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><msubsup><mi>I</mi> <mi>F</mi> <mo>•</mo></msubsup><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> 0 \to (R^n F(A)\coloneqq H^n(I^\bullet) \stackrel{\simeq}{\to} H^n(I_F^\bullet) \to 0 \,. </annotation></semantics></math></div></div> <h3 id="derived_adjoint_functors">Derived adjoint functors</h3> <div class="num_note"> <h6 id="observation">Observation</h6> <p>If</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>F</mi><mo>⊣</mo><mi>G</mi><mo stretchy="false">)</mo><mo>:</mo><mi>𝒜</mi><mover><munder><mo>→</mo><mi>F</mi></munder><mover><mo>←</mo><mi>G</mi></mover></mover><mi>ℬ</mi></mrow><annotation encoding="application/x-tex"> (F \dashv G) : \mathcal{A} \stackrel{\overset{G}{\leftarrow}}{\underset{F}{\to}} \mathcal{B} </annotation></semantics></math></div> <p>is a pair of <a class="existingWikiWord" href="/nlab/show/additive+functor">additive</a> <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a>, then</p> <ul> <li> <p>the <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/right+exact+functor">right exact</a>;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> <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> is <a class="existingWikiWord" href="/nlab/show/left+exact+functor">left exact</a>;</p> </li> </ul> </div> <p>(…)</p> <h3 id="preservation_of_further_limits_and_colimits">Preservation of further limits and colimits</h3> <div class="num_prop"> <h6 id="proposition_6">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>𝒜</mi><mo>→</mo><mi>ℬ</mi></mrow><annotation encoding="application/x-tex">F \colon \mathcal{A} \to \mathcal{B}</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/additive+functor">additive</a> <a class="existingWikiWord" href="/nlab/show/right+exact+functor">right exact functor</a> with codomain an <a class="existingWikiWord" href="/nlab/show/additive+and+abelian+categories">AB5-category</a>.</p> <p>Then the <a class="existingWikiWord" href="/nlab/show/left+derived+functors">left derived functors</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mi>n</mi></msub><mi>F</mi></mrow><annotation encoding="application/x-tex">L_n F</annotation></semantics></math> preserves <a class="existingWikiWord" href="/nlab/show/filtered+colimits">filtered colimits</a> precisely if filtered colimits of <a class="existingWikiWord" href="/nlab/show/projective+objects">projective objects</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> are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/acyclic+objects">acyclic objects</a>.</p> </div> <p>See <a href="http://mathoverflow.net/questions/97658/left-derived-functors-commute-with-filtered-colimits">here</a>.</p> <h2 id="examples">Examples</h2> <p>(…)</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/delta-functor">delta-functor</a></li> </ul> <h2 id="references">References</h2> <p>A standard textbook introduction is chapter 2 of</p> <ul> <li id="Weibel"><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>.</li> </ul> <p>A systematic discussion from the point of view of <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> and <a class="existingWikiWord" href="/nlab/show/derived+categories">derived categories</a> is in chapter 7 of</p> <ul> <li id="Schapira"> <p><a class="existingWikiWord" href="/nlab/show/Pierre+Schapira">Pierre Schapira</a>, <em>Categories and homological algebra</em> (2011) (<a href="http://people.math.jussieu.fr/~schapira/lectnotes/HomAl.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Masaki+Kashiwara">Masaki Kashiwara</a>, <a class="existingWikiWord" href="/nlab/show/Pierre+Schapira">Pierre Schapira</a>, section 13 of <em><a class="existingWikiWord" href="/nlab/show/Categories+and+Sheaves">Categories and Sheaves</a></em>, Grundlehren der Mathematischen Wissenschaften <strong>332</strong>, Springer (2006)</p> </li> </ul> <p>The above text is taken from</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/HAI">Introduction to Homological Algebra</a></em></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on February 6, 2024 at 16:50:26. 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