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exact couple in nLab
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<div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 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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/5449/#Item_3" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="homological_algebra">Homological algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a></strong></p> <p>(also <a class="existingWikiWord" href="/nlab/show/nonabelian+homological+algebra">nonabelian homological algebra</a>)</p> <p><em><a class="existingWikiWord" href="/schreiber/show/Introduction+to+Homological+Algebra">Introduction</a></em></p> <p><strong>Context</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+and+abelian+categories">additive and abelian categories</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ab-enriched+category">Ab-enriched category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-additive+category">pre-additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-abelian+category">pre-abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+category">Grothendieck category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaves">abelian sheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semi-abelian+category">semi-abelian category</a></p> </li> </ul> <p><strong>Basic definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/kernel">kernel</a>, <a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex">complex</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential">differential</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homology">homology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+map">chain map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+homotopy">chain homotopy</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+homology+and+cohomology">chain homology and cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+resolution">homological resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+on+a+simplicial+set">simplicial homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+homology">generalized homology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exact+sequence">exact sequence</a>,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a>, <a class="existingWikiWord" href="/nlab/show/long+exact+sequence">long exact sequence</a>, <a class="existingWikiWord" href="/nlab/show/split+exact+sequence">split exact sequence</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+object">injective object</a>, <a class="existingWikiWord" href="/nlab/show/projective+object">projective object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+resolution">injective resolution</a>, <a class="existingWikiWord" href="/nlab/show/projective+resolution">projective resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/flat+resolution">flat resolution</a></p> </li> </ul> </li> </ul> <p><strong>Stable homotopy theory notions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a>, <a class="existingWikiWord" href="/nlab/show/enhanced+triangulated+category">enhanced triangulated category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+model+category">stable model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pretriangulated+dg-category">pretriangulated dg-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E-category">A-∞-category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+chain+complexes">(∞,1)-category of chain complexes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a>, <a class="existingWikiWord" href="/nlab/show/derived+functor+in+homological+algebra">derived functor in homological algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Tor">Tor</a>, <a class="existingWikiWord" href="/nlab/show/Ext">Ext</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lim%5E1+and+Milnor+sequences">lim^1 and Milnor sequences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fr-code">fr-code</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> </li> </ul> <p><strong>Constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/double+complex">double complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Koszul-Tate+resolution">Koszul-Tate resolution</a>, <a class="existingWikiWord" href="/nlab/show/BRST-BV+complex">BRST-BV complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+filtered+complex">spectral sequence of a filtered complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+double+complex">spectral sequence of a double complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+spectral+sequence">Grothendieck spectral sequence</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Leray+spectral+sequence">Leray spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Serre+spectral+sequence">Serre spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild-Serre+spectral+sequence">Hochschild-Serre spectral sequence</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong>Lemmas</strong></p> <p><a class="existingWikiWord" href="/nlab/show/diagram+chasing">diagram chasing</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/3x3+lemma">3x3 lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/four+lemma">four lemma</a>, <a class="existingWikiWord" href="/nlab/show/five+lemma">five lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/snake+lemma">snake lemma</a>, <a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/horseshoe+lemma">horseshoe lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Baer%27s+criterion">Baer's criterion</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/Schanuel%27s+lemma">Schanuel's lemma</a></p> <p><strong>Homology theories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cyclic+homology">cyclic homology</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> / <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal</a>, <a class="existingWikiWord" href="/nlab/show/operadic+Dold-Kan+correspondence">operadic</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Moore+complex">Moore complex</a>, <a class="existingWikiWord" href="/nlab/show/Alexander-Whitney+map">Alexander-Whitney map</a>, <a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+map">Eilenberg-Zilber map</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+theorem">Eilenberg-Zilber theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cochain+on+a+simplicial+set">cochain on a simplicial set</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#exact_couples'>Exact couples</a></li> <li><a href='#SpectralSequencesFromExactCouples'>Spectral sequences from exact couples</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#ExactCoupleOfATowerOfFibrations'>Exact couple of a tower of (co)-fibrations</a></li> </ul> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p><em>Exact couples</em> are a way to encode data that makes a <a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a>, specially adapted to the case that the underlying filtering along which the spectral sequence proceeds is induced from a tower of <a class="existingWikiWord" href="/nlab/show/homotopy+fibers">homotopy fibers</a>, such as a <a class="existingWikiWord" href="/nlab/show/Postnikov+tower">Postnikov tower</a> or <a class="existingWikiWord" href="/nlab/show/Adams+tower">Adams tower</a> (see also at <em><a class="existingWikiWord" href="/nlab/show/Adams+spectral+sequence">Adams spectral sequence</a></em>).</p> <h2 id="definition">Definition</h2> <h3 id="exact_couples">Exact couples</h3> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>Given an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, an <strong><a class="existingWikiWord" href="/nlab/show/exact+couple">exact couple</a></strong> in <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">\mathcal{C}</annotation></semantics></math> is a cyclic <a class="existingWikiWord" href="/nlab/show/long+exact+sequence">long exact sequence</a> of three <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> among two <a class="existingWikiWord" href="/nlab/show/objects">objects</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>⋯</mi><mover><mo>⟶</mo><mi>k</mi></mover><mi>E</mi><mover><mo>→</mo><mi>j</mi></mover><mi>D</mi><mover><mo>⟶</mo><mi>φ</mi></mover><mi>D</mi><mover><mo>→</mo><mi>k</mi></mover><mi>E</mi><mover><mo>⟶</mo><mi>j</mi></mover><mi>⋯</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \cdots \stackrel{k}{\longrightarrow} E \overset{j}{\to} D \overset{\varphi}{\longrightarrow} D \overset{k}{\to} E \overset{j}{\longrightarrow} \cdots \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>This being cyclic, it is usually depicted as a triangle</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>D</mi></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mi>φ</mi></mover></mtd> <mtd></mtd> <mtd><mi>D</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>j</mi></mpadded></msub><mo>↖</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mi>k</mi></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>E</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ D && \stackrel{\varphi}{\longrightarrow} && D \\ & {}_{\mathllap{j}}\nwarrow && \swarrow_{\mathrlap{k}} \\ && E } </annotation></semantics></math></div></div> <p>The archetypical example from which this and the following definition draw their meaning is example <a class="maruku-ref" href="#ExactCoupleOfATower"></a> below.</p> <h3 id="SpectralSequencesFromExactCouples">Spectral sequences from exact couples</h3> <div class="num_defn" id="CohomologySpectralSequence"> <h6 id="definition_3">Definition</h6> <p>A <em>cohomology <a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msubsup><mi>E</mi> <mi>r</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup><mo>,</mo><msub><mi>d</mi> <mi>r</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{E_r^{p,q}, d_r\}</annotation></semantics></math> is</p> <ol> <li> <p>a sequence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msubsup><mi>E</mi> <mi>r</mi> <mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msubsup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{E_r^{\bullet,\bullet}\}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">r \in \mathbb{Z}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">r \geq 2</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/bigraded+object">bigraded</a> <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a>;</p> </li> <li> <p>a sequence of <a class="existingWikiWord" href="/nlab/show/differentials">differentials</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>d</mi> <mi>r</mi></msub><mo lspace="verythinmathspace">:</mo><msubsup><mi>E</mi> <mi>r</mi> <mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msubsup><mo>⟶</mo><msubsup><mi>E</mi> <mi>r</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>r</mi><mo>,</mo><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{d_r \colon E_r^{\bullet,\bullet} \longrightarrow E_r^{\bullet+r, \bullet-r+1}\}</annotation></semantics></math></p> </li> </ol> <p>such that</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow> <mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">H_{r+1}^{\bullet,\bullet}</annotation></semantics></math> is 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><msub><mi>d</mi> <mi>r</mi></msub></mrow><annotation encoding="application/x-tex">d_r</annotation></semantics></math>:, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>E</mi> <mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow> <mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msubsup><mo>=</mo><mi>H</mi><mo stretchy="false">(</mo><msubsup><mi>E</mi> <mi>r</mi> <mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msubsup><mo>,</mo><msub><mi>d</mi> <mi>r</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E_{r+1}^{\bullet, \bullet} = H(E_r^{\bullet,\bullet},d_r)</annotation></semantics></math>.</li> </ul> <p>Given a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/graded+abelian+group">graded abelian group</a>_ <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">C^\bullet</annotation></semantics></math> equipped with a decreasing <a class="existingWikiWord" href="/nlab/show/filtration">filtration</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>•</mo></msup><mo>⊃</mo><mi>⋯</mi><mo>⊃</mo><msup><mi>F</mi> <mi>s</mi></msup><msup><mi>C</mi> <mo>•</mo></msup><mo>⊃</mo><msup><mi>F</mi> <mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msup><msup><mi>C</mi> <mo>•</mo></msup><mo>⊃</mo><mi>⋯</mi><mo>⊃</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> C^\bullet \supset \cdots \supset F^s C^\bullet \supset F^{s+1} C^\bullet \supset \cdots \supset 0 </annotation></semantics></math></div> <p>such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>•</mo></msup><mo>=</mo><munder><mo>∪</mo><mi>s</mi></munder><msup><mi>F</mi> <mi>s</mi></msup><msup><mi>C</mi> <mo>•</mo></msup><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>and</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mn>0</mn><mo>=</mo><munder><mo>∩</mo><mi>s</mi></munder><msup><mi>F</mi> <mi>s</mi></msup><msup><mi>C</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex"> C^\bullet = \underset{s}{\cup} F^s C^\bullet \;\;\;\; and \;\;\;\; 0 = \underset{s}{\cap} F^s C^\bullet </annotation></semantics></math></div> <p>then the spectral sequence is said to <em>converge</em> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">C^\bullet</annotation></semantics></math>, denoted,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>E</mi> <mn>2</mn> <mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msubsup><mo>⇒</mo><msup><mi>C</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex"> E_2^{\bullet,\bullet} \Rightarrow C^\bullet </annotation></semantics></math></div> <p>if</p> <ol> <li> <p>in each bidegree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(s,t)</annotation></semantics></math> the sequence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msubsup><mi>E</mi> <mi>r</mi> <mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msubsup><msub><mo stretchy="false">}</mo> <mi>r</mi></msub></mrow><annotation encoding="application/x-tex">\{E_r^{s,t}\}_r</annotation></semantics></math> eventually becomes constant on a group</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>E</mi> <mn>∞</mn> <mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msubsup><mo>≔</mo><msubsup><mi>E</mi> <mrow><mo>≫</mo><mn>1</mn></mrow> <mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">E_\infty^{s,t} \coloneqq E_{\gg 1}^{s,t}</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>E</mi> <mn>∞</mn> <mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">E_\infty^{\bullet,\bullet}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/associated+graded">associated graded</a> of the filtered <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">C^\bullet</annotation></semantics></math> in that</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>E</mi> <mn>∞</mn> <mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msubsup><mo>≃</mo><msup><mi>F</mi> <mi>s</mi></msup><msup><mi>C</mi> <mrow><mi>s</mi><mo>+</mo><mi>t</mi></mrow></msup><mo stretchy="false">/</mo><msup><mi>F</mi> <mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msup><msup><mi>C</mi> <mrow><mi>s</mi><mo>+</mo><mi>t</mi></mrow></msup></mrow><annotation encoding="application/x-tex">E_\infty^{s,t} \simeq F^s C^{s+t} / F^{s+1}C^{s+t}</annotation></semantics></math>.</p> </li> </ol> <p>The converging spectral sequence is called <em>multiplicative</em> if</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msubsup><mi>E</mi> <mn>2</mn> <mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msubsup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{E_2^{\bullet,\bullet}\}</annotation></semantics></math> is equipped with the structure of a <a class="existingWikiWord" href="/nlab/show/bigraded+object">bigraded object</a> <a class="existingWikiWord" href="/nlab/show/associative+algebra">algebra</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>F</mi> <mo>•</mo></msup><msup><mi>C</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">F^\bullet C^\bullet</annotation></semantics></math> is equipped with the structure of a filtered <a class="existingWikiWord" href="/nlab/show/graded+algebra">graded algebra</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>F</mi> <mi>p</mi></msup><msup><mi>C</mi> <mi>k</mi></msup><mo>⋅</mo><msup><mi>F</mi> <mi>q</mi></msup><msup><mi>C</mi> <mi>l</mi></msup><mo>⊂</mo><msup><mi>F</mi> <mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></msup><msup><mi>C</mi> <mrow><mi>k</mi><mo>+</mo><mi>l</mi></mrow></msup></mrow><annotation encoding="application/x-tex">F^p C^k \cdot F^q C^l \subset F^{p+q} C^{k+l}</annotation></semantics></math>);</p> </li> </ol> <p>such that</p> <ol> <li> <p>each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>r</mi></msub></mrow><annotation encoding="application/x-tex">d_{r}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/derivation">derivation</a> with respect to the (induced) algebra structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><msubsup><mi>E</mi> <mi>r</mi> <mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msubsup></mrow></mrow><annotation encoding="application/x-tex">{E_r^{\bullet,\bullet}}</annotation></semantics></math>, graded of degree 1 with respect to total degree;</p> </li> <li> <p>the multiplication on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>E</mi> <mn>∞</mn> <mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">E_\infty^{\bullet,\bullet}</annotation></semantics></math> is compatible with that on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">C^\bullet</annotation></semantics></math>.</p> </li> </ol> </div> <div class="num_defn" id="ExactCoupleAndDerivedExactCouple"> <h6 id="definition_4">Definition</h6> <p><strong>(derived exact couples)</strong></p> <p>An <em><a class="existingWikiWord" href="/nlab/show/exact+couple">exact couple</a></em> is three <a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a> of <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>D</mi></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mi>g</mi></mover></mtd> <mtd></mtd> <mtd><mi>D</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msub><mo>↖</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mi>h</mi></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>E</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ D && \stackrel{g}{\longrightarrow} && D \\ & {}_{\mathllap{f}}\nwarrow && \swarrow_{\mathrlap{h}} \\ && E } </annotation></semantics></math></div> <p>such that the <a class="existingWikiWord" href="/nlab/show/image">image</a> of one is the <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> of the next.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>im</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ker</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>im</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ker</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>im</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ker</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> im(h) = ker(f)\,,\;\;\; im(f) = ker(g)\,, \;\;\; im(g) = ker(h) \,. </annotation></semantics></math></div> <p>Given an exact couple, then its <em>derived exact couple</em> is</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>im</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mi>g</mi></mover></mtd> <mtd></mtd> <mtd><mi>im</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msub><mo>↖</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><mi>h</mi><mo>∘</mo><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>H</mi><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mi>h</mi><mo>∘</mo><mi>f</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ im(g) && \stackrel{g}{\longrightarrow} && im(g) \\ & {}_{\mathllap{f}}\nwarrow && \swarrow_{\mathrlap{h \circ g^{-1}}} \\ && H(E, h \circ f) } \,. </annotation></semantics></math></div></div> <p>Here and in the following we write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">g^{-1}</annotation></semantics></math> etc. for the operation of choosing a <a class="existingWikiWord" href="/nlab/show/preimage">preimage</a> under a given function <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>. In each case it is left implicit that the given expression is independent of which choice is made.</p> <div class="num_prop" id="CohomologicalSpectralSequenceOfAnExactCouple"> <h6 id="proposition">Proposition</h6> <p><strong>(cohomological spectral sequence of an exact couple)</strong></p> <p>Given an exact couple, def. <a class="maruku-ref" href="#ExactCoupleAndDerivedExactCouple"></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><msub><mi>D</mi> <mn>1</mn></msub></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>g</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd></mtd> <mtd><msub><mi>D</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow></mpadded></msub><mo>↖</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><msub><mi>h</mi> <mn>1</mn></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>E</mi> <mn>1</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ D_1 && \stackrel{g_1}{\longrightarrow} && D_1 \\ & {}_{\mathllap{f_1}}\nwarrow && \swarrow_{\mathrlap{h_1}} \\ && E_1 } </annotation></semantics></math></div> <p>its derived exact couple</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>D</mi> <mn>2</mn></msub></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>g</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd></mtd> <mtd><msub><mi>D</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>f</mi> <mn>2</mn></msub></mrow></mpadded></msub><mo>↖</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><msub><mi>h</mi> <mn>2</mn></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>E</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ D_2 && \stackrel{g_2}{\longrightarrow} && D_2 \\ & {}_{\mathllap{f_2}}\nwarrow && \swarrow_{\mathrlap{h_2}} \\ && E_2 } </annotation></semantics></math></div> <p>is itself an exact couple. Accordingly there is induced a sequence of exact couples</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>D</mi> <mi>r</mi></msub></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>g</mi> <mi>r</mi></msub></mrow></mover></mtd> <mtd></mtd> <mtd><msub><mi>D</mi> <mi>r</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>f</mi> <mi>r</mi></msub></mrow></mpadded></msub><mo>↖</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><msub><mi>h</mi> <mi>r</mi></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>E</mi> <mi>r</mi></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ D_r && \stackrel{g_r}{\longrightarrow} && D_r \\ & {}_{\mathllap{f_r}}\nwarrow && \swarrow_{\mathrlap{h_r}} \\ && E_r } \,. </annotation></semantics></math></div> <p>If the abelian groups <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> are equipped with <a class="existingWikiWord" href="/nlab/show/bigraded+object">bigrading</a> such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>deg</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>deg</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>deg</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> deg(f) = (0,0)\,,\;\;\;\; deg(g) = (-1,1)\,,\;\;\; deg(h) = (1,0) </annotation></semantics></math></div> <p>then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msubsup><mi>E</mi> <mi>r</mi> <mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msubsup><mo>,</mo><msub><mi>d</mi> <mi>r</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{E_r^{\bullet,\bullet}, d_r\}</annotation></semantics></math> with</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><msub><mi>d</mi> <mi>r</mi></msub></mtd> <mtd><mo>≔</mo><msub><mi>h</mi> <mi>r</mi></msub><mo>∘</mo><msub><mi>f</mi> <mi>r</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>h</mi><mo>∘</mo><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>∘</mo><mi>f</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} d_r & \coloneqq h_r \circ f_r \\ & = h \circ g^{-r+1} \circ f \end{aligned} </annotation></semantics></math></div> <p>is a cohomological spectral sequence, def. <a class="maruku-ref" href="#CohomologySpectralSequence"></a>.</p> <p>(As before in prop. <a class="maruku-ref" href="#CohomologicalSpectralSequenceOfAnExactCouple"></a>, the notation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi></mrow></msup></mrow><annotation encoding="application/x-tex">g^{-n}</annotation></semantics></math> with <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> denotes the function given by choosing, on representatives, a <a class="existingWikiWord" href="/nlab/show/preimage">preimage</a> under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>g</mi> <mi>n</mi></msup><mo>=</mo><munder><munder><mrow><mi>g</mi><mo>∘</mo><mi>⋯</mi><mo>∘</mo><mi>g</mi><mo>∘</mo><mi>g</mi></mrow><mo>⏟</mo></munder><mrow><mi>n</mi><mspace width="thickmathspace"></mspace><mi>times</mi></mrow></munder></mrow><annotation encoding="application/x-tex">g^n = \underset{n\;times}{\underbrace{g \circ \cdots \circ g \circ g}}</annotation></semantics></math>, with the implicit claim that all possible choices represent the same equivalence class.)</p> <p>If for every bidegree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(s,t)</annotation></semantics></math> there exists <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>R</mi> <mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>≫</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">R_{s,t} \gg 1</annotation></semantics></math> such that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>≥</mo><msub><mi>R</mi> <mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow><annotation encoding="application/x-tex">r \geq R_{s,t}</annotation></semantics></math></p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo lspace="verythinmathspace">:</mo><msup><mi>D</mi> <mrow><mi>s</mi><mo>+</mo><mi>R</mi><mo>,</mo><mi>t</mi><mo>−</mo><mi>R</mi></mrow></msup><mover><mo>⟶</mo><mo>≃</mo></mover><msup><mi>D</mi> <mrow><mi>s</mi><mo>+</mo><mi>R</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>t</mi><mo>−</mo><mi>R</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">g \colon D^{s+R,t-R} \stackrel {\simeq}{\longrightarrow} D^{s+R -1, t-R-1}</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo lspace="verythinmathspace">:</mo><msup><mi>D</mi> <mrow><mi>s</mi><mo>−</mo><mi>R</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>t</mi><mo>+</mo><mi>R</mi><mo>−</mo><mn>2</mn></mrow></msup><mover><mo>⟶</mo><mn>0</mn></mover><msup><mi>D</mi> <mrow><mi>s</mi><mo>−</mo><mi>R</mi><mo>,</mo><mi>t</mi><mo>+</mo><mi>R</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">g\colon D^{s-R+1, t+R-2} \stackrel{0}{\longrightarrow} D^{s-R,t+R-1}</annotation></semantics></math></p> </li> </ol> <p>then this spectral sequence converges to the <a class="existingWikiWord" href="/nlab/show/inverse+limit">inverse limit</a> group</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>G</mi> <mo>•</mo></msup><mo>≔</mo><munder><mi>lim</mi><mrow></mrow></munder><mrow><mo>(</mo><mi>⋯</mi><mover><mo>→</mo><mi>g</mi></mover><msup><mi>D</mi> <mrow><mi>s</mi><mo>,</mo><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>s</mi></mrow></msup><mover><mo>⟶</mo><mi>g</mi></mover><msup><mi>D</mi> <mrow><mi>s</mi><mo>−</mo><mn>1</mn><mo>,</mo><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msup><mover><mo>→</mo><mi>g</mi></mover><mi>⋯</mi><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> G^\bullet \coloneqq \underset{}{\lim} \left( \cdots \stackrel{g}{\to} D^{s,\bullet-s} \stackrel{g}{\longrightarrow} D^{s-1, \bullet - s + 1} \stackrel{g}{\to} \cdots \right) </annotation></semantics></math></div> <p>filtered by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>F</mi> <mi>p</mi></msup><msup><mi>G</mi> <mo>•</mo></msup><mo>≔</mo><mi>ker</mi><mo stretchy="false">(</mo><msup><mi>G</mi> <mo>•</mo></msup><mo>→</mo><msup><mi>D</mi> <mrow><mi>p</mi><mo>−</mo><mn>1</mn><mo>,</mo><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> F^p G^\bullet \coloneqq ker(G^\bullet \to D^{p-1, \bullet - p+1}) \,. </annotation></semantics></math></div></div> <p>(e.g. <a href="#Kochmann96">Kochmann 96, lemma 2.6.2</a>)</p> <div class="proof"> <h6 id="proof">Proof</h6> <p>We check the claimed form of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">E_\infty</annotation></semantics></math>-page:</p> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ker</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mo>=</mo><mi>im</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ker(h) = im(g)</annotation></semantics></math> in the exact couple, the kernel</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ker</mi><mo stretchy="false">(</mo><msub><mi>d</mi> <mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>≔</mo><mi>ker</mi><mo stretchy="false">(</mo><mi>h</mi><mo>∘</mo><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>r</mi><mo>+</mo><mn>2</mn></mrow></msup><mo>∘</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> ker(d_{r-1}) \coloneqq ker(h \circ g^{-r+2} \circ f) </annotation></semantics></math></div> <p>consists of those elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>r</mi><mo>+</mo><mn>2</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g^{-r+2} (f(x)) = g(y)</annotation></semantics></math>, for some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math>, hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ker</mi><mo stretchy="false">(</mo><msub><mi>d</mi> <mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msub><msup><mo stretchy="false">)</mo> <mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msup><mo>≃</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msup><mi>g</mi> <mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msup><mi>D</mi> <mrow><mi>s</mi><mo>+</mo><mi>r</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>t</mi><mo>−</mo><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> ker(d_{r-1})^{s,t} \simeq f^{-1}(g^{r-1}(D^{s+r-1,t-r+1})) \,. </annotation></semantics></math></div> <p>By assumption there is for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(s,t)</annotation></semantics></math> an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>R</mi> <mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow><annotation encoding="application/x-tex">R_{s,t}</annotation></semantics></math> such that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>≥</mo><msub><mi>R</mi> <mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow><annotation encoding="application/x-tex">r \geq R_{s,t}</annotation></semantics></math> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ker</mi><mo stretchy="false">(</mo><msub><mi>d</mi> <mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msub><msup><mo stretchy="false">)</mo> <mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msup></mrow><annotation encoding="application/x-tex">ker(d_{r-1})^{s,t}</annotation></semantics></math> is independent of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math>.</p> <p>Moreover, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>im</mi><mo stretchy="false">(</mo><msub><mi>d</mi> <mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">im(d_{r-1})</annotation></semantics></math> consists of the image under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math> of those <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><msup><mi>D</mi> <mrow><mi>s</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>t</mi></mrow></msup></mrow><annotation encoding="application/x-tex">x \in D^{s-1,t}</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>g</mi> <mrow><mi>r</mi><mo>−</mo><mn>2</mn></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g^{r-2}(x)</annotation></semantics></math> is in the image 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>, hence (since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>im</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ker</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">im(f) = ker(g)</annotation></semantics></math> by exactness of the exact couple) such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>g</mi> <mrow><mi>r</mi><mo>−</mo><mn>2</mn></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g^{r-2}(x)</annotation></semantics></math> is in the kernel of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math>, hence such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> is in the kernel of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>g</mi> <mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">g^{r-1}</annotation></semantics></math>. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>></mo><mi>R</mi></mrow><annotation encoding="application/x-tex">r \gt R</annotation></semantics></math> then by assumption <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>g</mi> <mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msup><msub><mo stretchy="false">|</mo> <mrow><msup><mi>D</mi> <mrow><mi>s</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>t</mi></mrow></msup></mrow></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">g^{r-1}|_{D^{s-1,t}} = 0</annotation></semantics></math> and so then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>im</mi><mo stretchy="false">(</mo><msub><mi>d</mi> <mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mi>im</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">im(d_{r-1}) = im(h)</annotation></semantics></math>.</p> <p>(Beware this subtlety: while <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>g</mi> <mrow><msub><mi>R</mi> <mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></msup><msub><mo stretchy="false">|</mo> <mrow><msup><mi>D</mi> <mrow><mi>s</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>t</mi></mrow></msup></mrow></msub></mrow><annotation encoding="application/x-tex">g^{R_{s,t}}|_{D^{s-1,t}}</annotation></semantics></math> vanishes by the convergence assumption, the expression <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>g</mi> <mrow><msub><mi>R</mi> <mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow></msup><msub><mo stretchy="false">|</mo> <mrow><msup><mi>D</mi> <mrow><mi>s</mi><mo>+</mo><mi>r</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>t</mi><mo>−</mo><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></msub></mrow><annotation encoding="application/x-tex">g^{R_{s,t}}|_{D^{s+r-1,t-r+1}}</annotation></semantics></math> need not vanish yet. Only the higher power <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>g</mi> <mrow><msub><mi>R</mi> <mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>+</mo><msub><mi>R</mi> <mrow><mi>s</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>t</mi><mo>+</mo><mn>2</mn></mrow></msub><mo>+</mo><mn>2</mn></mrow></msup><msub><mo stretchy="false">|</mo> <mrow><msup><mi>D</mi> <mrow><mi>s</mi><mo>+</mo><mi>r</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>t</mi><mo>−</mo><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></msub></mrow><annotation encoding="application/x-tex">g^{R_{s,t}+ R_{s+1,t+2}+2}|_{D^{s+r-1,t-r+1}}</annotation></semantics></math> is again guaranteed to vanish. )</p> <p id="InfinityPage"> It follows that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msubsup><mi>E</mi> <mn>∞</mn> <mrow><mi>p</mi><mo>,</mo><mi>n</mi><mo>−</mo><mi>p</mi></mrow></msubsup></mtd> <mtd><mo>=</mo><mi>ker</mi><mo stretchy="false">(</mo><msub><mi>d</mi> <mi>R</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>im</mi><mo stretchy="false">(</mo><msub><mi>d</mi> <mi>R</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>im</mi><mo stretchy="false">(</mo><msup><mi>g</mi> <mrow><mi>R</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>im</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>im</mi><mo stretchy="false">(</mo><msup><mi>g</mi> <mrow><mi>R</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>ker</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><munderover><mo>⟶</mo><mo>≃</mo><mi>f</mi></munderover><mi>im</mi><mo stretchy="false">(</mo><msup><mi>g</mi> <mrow><mi>R</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>∩</mo><mi>im</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>im</mi><mo stretchy="false">(</mo><msup><mi>g</mi> <mrow><mi>R</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>∩</mo><mi>ker</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} E_\infty^{p,n-p} & = ker(d_R)/im(d_R) \\ & \simeq f^{-1}(im(g^{R-1}))/im(h) \\ & \simeq f^{-1}(im(g^{R-1}))/ker(f) \\ & \underoverset{\simeq}{f}{\longrightarrow} im(g^{R-1}) \cap im(f) \\ & \simeq im(g^{R-1}) \cap ker(g) \end{aligned} </annotation></semantics></math></div> <p>where in last two steps we used once more the exactness of the exact couple.</p> <p id="InfinityPageIsSubgroupOfImageOfFirstPageUnderf">(Notice that the above equation means in particular that the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">E_\infty</annotation></semantics></math>-page is a sub-group of the image of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">E_1</annotation></semantics></math>-page under <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>.)</p> <p>The last group above is that of elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><msup><mi>G</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">x \in G^n</annotation></semantics></math> which map to zero in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mrow><mi>p</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>−</mo><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">D^{p-1,n-p+1}</annotation></semantics></math> and where two such are identified if they agree in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mrow><mi>p</mi><mo>,</mo><mi>n</mi><mo>−</mo><mi>p</mi></mrow></msup></mrow><annotation encoding="application/x-tex">D^{p,n-p}</annotation></semantics></math>, hence indeed</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>E</mi> <mn>∞</mn> <mrow><mi>p</mi><mo>,</mo><mi>n</mi><mo>−</mo><mi>p</mi></mrow></msubsup><mo>≃</mo><msup><mi>F</mi> <mi>p</mi></msup><msup><mi>G</mi> <mi>n</mi></msup><mo stretchy="false">/</mo><msup><mi>F</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><msup><mi>G</mi> <mi>n</mi></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> E_\infty^{p,n-p} \simeq F^p G^n / F^{p+1} G^n \,. </annotation></semantics></math></div></div> <h2 id="examples">Examples</h2> <h3 id="ExactCoupleOfATowerOfFibrations">Exact couple of a tower of (co)-fibrations</h3> <p>…<a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+tower+of+fibrations">spectral sequence of a tower of fibrations</a>…</p> <div class="num_defn" id="FilteredSpectrum"> <h6 id="definition_5">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/filtered+spectrum">filtered spectrum</a> is a <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> equipped with a sequence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><mi>ℕ</mi><mo>,</mo><mo>></mo><mo stretchy="false">)</mo><mo>⟶</mo><mi>Spectra</mi></mrow><annotation encoding="application/x-tex">X_\bullet \colon (\mathbb{N}, \gt) \longrightarrow Spectra</annotation></semantics></math> of spectra 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>⋯</mi><mo>⟶</mo><msub><mi>X</mi> <mn>3</mn></msub><mover><mo>⟶</mo><mrow><msub><mi>f</mi> <mn>2</mn></msub></mrow></mover><msub><mi>X</mi> <mn>2</mn></msub><mover><mo>⟶</mo><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow></mover><msub><mi>X</mi> <mn>1</mn></msub><mover><mo>⟶</mo><mrow><msub><mi>f</mi> <mn>0</mn></msub></mrow></mover><msub><mi>X</mi> <mn>0</mn></msub><mo>=</mo><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \cdots \longrightarrow X_3 \stackrel{f_2}{\longrightarrow} X_2 \stackrel{f_1}{\longrightarrow} X_1 \stackrel{f_0}{\longrightarrow} X_0 = X \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>More generally a <a class="existingWikiWord" href="/nlab/show/filtered+object+in+an+%28infinity%2C1%29-category">filtering</a> on an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> in (stable or not) <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-graded sequence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">X_\bullet </annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>≃</mo><munder><mi>lim</mi><mo>⟶</mo></munder><msub><mi>X</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">X\simeq \underset{\longrightarrow}{\lim} X_\bullet</annotation></semantics></math>. But for the present purpose we stick with the simpler special case of def. <a class="maruku-ref" href="#FilteredSpectrum"></a>.</p> </div> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>There is <em>no</em> condition on the <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> in def. <a class="maruku-ref" href="#FilteredSpectrum"></a>. In particular, they are <em>not</em> required to be <a class="existingWikiWord" href="/nlab/show/n-monomorphisms">n-monomorphisms</a> or <a class="existingWikiWord" href="/nlab/show/n-epimorphisms">n-epimorphisms</a> for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>.</p> <p>On the other hand, while they are also not explicitly required to have a presentation by <a class="existingWikiWord" href="/nlab/show/cofibrations">cofibrations</a> or <a class="existingWikiWord" href="/nlab/show/fibrations">fibrations</a>, this follows automatically: by the existence of <a class="existingWikiWord" href="/nlab/show/model+structures+for+spectra">model structures for spectra</a>, every filtering on a spectrum is equivalent to one in which all morphisms are represented by <a class="existingWikiWord" href="/nlab/show/cofibrations">cofibrations</a> or by <a class="existingWikiWord" href="/nlab/show/fibrations">fibrations</a>.</p> <p>This means that we may think of a filtration on a spectrum <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> in the sense of def. <a class="maruku-ref" href="#FilteredSpectrum"></a> as equivalently being a <a class="existingWikiWord" href="/nlab/show/tower+of+fibrations">tower of fibrations</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> </div> <p>The following remark <a class="maruku-ref" href="#UnrolledExactCoupleOfAFiltrationOnASpectrum"></a> unravels the structure encoded in a filtration on a spectrum, and motivates the concepts of <a class="existingWikiWord" href="/nlab/show/exact+couples">exact couples</a> and their <a class="existingWikiWord" href="/nlab/show/spectral+sequences">spectral sequences</a> from these.</p> <div class="num_remark" id="UnrolledExactCoupleOfAFiltrationOnASpectrum"> <h6 id="remark_4">Remark</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/filtered+spectrum">filtered spectrum</a> as in def. <a class="maruku-ref" href="#FilteredSpectrum"></a>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">A_k</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/homotopy+cofiber">homotopy cofiber</a> of its <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>th stage, such as to obtain the 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><mi>⋯</mi></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><msub><mi>X</mi> <mn>3</mn></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>f</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msub><mi>X</mi> <mn>2</mn></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>f</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msub><mi>X</mi> <mn>1</mn></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>A</mi> <mn>3</mn></msub></mtd> <mtd></mtd> <mtd><msub><mi>A</mi> <mn>2</mn></msub></mtd> <mtd></mtd> <mtd><msub><mi>A</mi> <mn>1</mn></msub></mtd> <mtd></mtd> <mtd><msub><mi>A</mi> <mn>0</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \cdots &\stackrel{}{\longrightarrow}& X_3 &\stackrel{f_2}{\longrightarrow}& X_2 &\stackrel{f_2}{\longrightarrow} & X_1 &\stackrel{f_1}{\longrightarrow}& X \\ && \downarrow && \downarrow && \downarrow && \downarrow \\ && A_3 && A_2 && A_1 && A_0 } </annotation></semantics></math></div> <p>where each stage</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>X</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>f</mi> <mi>k</mi></msub></mrow></mover></mtd> <mtd><msub><mi>X</mi> <mi>k</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>cofib</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>k</mi></msub><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>A</mi> <mi>k</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X_{k+1} &\stackrel{f_k}{\longrightarrow}& X_k \\ && \downarrow^{\mathrlap{cofib(f_k)}} \\ && A_k } </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequence">homotopy fiber sequence</a>.</p> <p>To break this down into invariants, apply the <a class="existingWikiWord" href="/nlab/show/stable+homotopy+groups">stable homotopy groups</a>-<a class="existingWikiWord" href="/nlab/show/functor">functor</a>. This yields a diagram of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/graded+abelian+groups">graded abelian groups</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>⋯</mi></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>3</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>3</mn></msub><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \cdots &\stackrel{}{\longrightarrow}& \pi_\bullet(X_3) &\stackrel{\pi_\bullet(f_2)}{\longrightarrow}& \pi_\bullet(X_2) &\stackrel{\pi_\bullet(f_2)}{\longrightarrow} & \pi_\bullet(X_1) &\stackrel{\pi_\bullet(f_1)}{\longrightarrow}& \pi_\bullet(X_0) \\ && \downarrow && \downarrow && \downarrow && \downarrow \\ && \pi_\bullet(A_3) && \pi_\bullet(A_2) && \pi_\bullet(A_1) && \pi_\bullet(A_0) } \,. </annotation></semantics></math></div> <p>Here each hook at stage <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> extends to a <a class="existingWikiWord" href="/nlab/show/long+exact+sequence+of+homotopy+groups">long exact sequence of homotopy groups</a> via <a class="existingWikiWord" href="/nlab/show/connecting+homomorphisms">connecting homomorphisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>δ</mi> <mo>•</mo> <mi>k</mi></msubsup></mrow><annotation encoding="application/x-tex">\delta_\bullet^k</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>→</mo><msub><mi>π</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msubsup><mi>δ</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow> <mi>k</mi></msubsup></mrow></mover><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>k</mi></msub><mo stretchy="false">)</mo></mrow></mover><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow></mrow></mover><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msubsup><mi>δ</mi> <mo>•</mo> <mi>k</mi></msubsup></mrow></mover><msub><mi>π</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>→</mo><mi>⋯</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \cdots \to \pi_{\bullet+1}(A_k) \stackrel{\delta_{\bullet+1}^k}{\longrightarrow} \pi_\bullet(X_{k+1}) \stackrel{\pi_\bullet(f_k)}{\longrightarrow} \pi_\bullet(X_k) \stackrel{}{\longrightarrow} \pi_\bullet(A_k) \stackrel{\delta_\bullet^k}{\longrightarrow} \pi_{\bullet-1}(X_{k+1}) \to \cdots \,. </annotation></semantics></math></div> <p>If we understand the <a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>δ</mi> <mi>k</mi></msub><mo lspace="verythinmathspace">:</mo><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \delta_k \colon \pi_\bullet(A_k) \longrightarrow \pi_\bullet(X_{k+1}) </annotation></semantics></math></div> <p>as a morphism of degree -1, then all this information fits into one diagram of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>⋯</mi></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>3</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>δ</mi> <mn>2</mn></msub></mrow></mpadded></msub><mo>↖</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>δ</mi> <mn>1</mn></msub></mrow></mpadded></msub><mo>↖</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>δ</mi> <mn>0</mn></msub></mrow></mpadded></msub><mo>↖</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>3</mn></msub><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ \cdots &\stackrel{}{\longrightarrow}& \pi_\bullet(X_3) &\stackrel{\pi_\bullet(f_2)}{\longrightarrow}& \pi_\bullet(X_2) &\stackrel{\pi_\bullet(f_2)}{\longrightarrow} & \pi_\bullet(X_1) &\stackrel{\pi_\bullet(f_1)}{\longrightarrow}& \pi_\bullet(X_0) \\ && \downarrow &{}_{\mathllap{\delta_2}}\nwarrow & \downarrow &{}_{\mathllap{\delta_1}}\nwarrow & \downarrow &{}_{\mathllap{\delta_0}}\nwarrow & \downarrow \\ && \pi_\bullet(A_3) && \pi_\bullet(A_2) && \pi_\bullet(A_1) && \pi_\bullet(A_0) } \,, </annotation></semantics></math></div> <p>where each triangle is a rolled-up incarnation of a <a class="existingWikiWord" href="/nlab/show/long+exact+sequence+of+homotopy+groups">long exact sequence of homotopy groups</a> (and in particular is <em>not</em> a commuting diagram!).</p> <p>If we furthermore consider the <a class="existingWikiWord" href="/nlab/show/bigraded+object">bigraded</a> <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mo>•</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_\bullet(X_\bullet)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mo>•</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_\bullet(A_\bullet)</annotation></semantics></math>, then this information may further be rolled-up to a single diagram of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mo>•</mo></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>f</mi> <mo>•</mo></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mo>•</mo></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>δ</mi></mpadded></msub><mo>↖</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>cofib</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mo>•</mo></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \pi_\bullet(X_\bullet) & \stackrel{\pi_\bullet(f_\bullet)}{\longrightarrow} & \pi_\bullet(X_\bullet) \\ & {}_{\mathllap{\delta}}\nwarrow & \downarrow^{\mathrlap{\pi_\bullet(cofib(f_\bullet))}} \\ && \pi_\bullet(A_\bullet) } </annotation></semantics></math></div> <p>where the morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>f</mi> <mo>•</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_\bullet(f_\bullet)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>cofib</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_\bullet(cofib(f_\bullet))</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>δ</mi></mrow><annotation encoding="application/x-tex">\delta</annotation></semantics></math> have bi-degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,-1)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,0)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-1,1)</annotation></semantics></math>, respectively.</p> <p>Here it is convenient to shift the bigrading, equivalently, by setting</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> <mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msup><mo>≔</mo><msub><mi>π</mi> <mrow><mi>t</mi><mo>−</mo><mi>s</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>s</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathcal{D}^{s,t} \coloneqq \pi_{t-s}(X_s) </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℰ</mi> <mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msup><mo>≔</mo><msub><mi>π</mi> <mrow><mi>t</mi><mo>−</mo><mi>s</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>s</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathcal{E}^{s,t} \coloneqq \pi_{t-s}(A_s) \,, </annotation></semantics></math></div> <p>because then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math> counts the cycles of going around the triangles:</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>𝒟</mi> <mrow><mi>s</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msup><mover><mo>⟶</mo><mrow><msub><mi>π</mi> <mrow><mi>t</mi><mo>−</mo><mi>s</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>s</mi></msub><mo stretchy="false">)</mo></mrow></mover><msup><mi>𝒟</mi> <mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msup><mover><mo>⟶</mo><mrow><msub><mi>π</mi> <mrow><mi>t</mi><mo>−</mo><mi>s</mi></mrow></msub><mo stretchy="false">(</mo><mi>cofib</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>s</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mover><msup><mi>ℰ</mi> <mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msup><mover><mo>⟶</mo><mrow><msub><mi>δ</mi> <mi>s</mi></msub></mrow></mover><msup><mi>𝒟</mi> <mrow><mi>s</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>t</mi></mrow></msup><mo>→</mo><mi>⋯</mi></mrow><annotation encoding="application/x-tex"> \cdots \to \mathcal{D}^{s+1,t+1} \stackrel{\pi_{t-s}(f_s)}{\longrightarrow} \mathcal{D}^{s,t} \stackrel{\pi_{t-s}(cofib(f_s))}{\longrightarrow} \mathcal{E}^{s,t} \stackrel{\delta_s}{\longrightarrow} \mathcal{D}^{s+1,t} \to \cdots </annotation></semantics></math></div> <p>Data of this form is called an <em><a class="existingWikiWord" href="/nlab/show/exact+couple">exact couple</a></em>, def. <a class="maruku-ref" href="#ExactCouple"></a> below.</p> </div> <div class="num_defn" id="UnrolledExactCouple"> <h6 id="definition_6">Definition</h6> <p>An <em>unrolled <a class="existingWikiWord" href="/nlab/show/exact+couple">exact couple</a></em> (of Adams-type) is a diagram of <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>⋯</mi></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><msup><mi>𝒟</mi> <mrow><mn>3</mn><mo>,</mo><mo>•</mo></mrow></msup></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>i</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msup><mi>𝒟</mi> <mrow><mn>2</mn><mo>,</mo><mo>•</mo></mrow></msup></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><msup><mi>𝒟</mi> <mrow><mn>1</mn><mo>,</mo><mo>•</mo></mrow></msup></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></mover></mtd> <mtd><msup><mi>𝒟</mi> <mrow><mn>0</mn><mo>,</mo><mo>•</mo></mrow></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow></mrow></mpadded></msup></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>k</mi> <mn>2</mn></msub></mrow></mpadded></msub><mo>↖</mo></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>j</mi> <mn>2</mn></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>k</mi> <mn>1</mn></msub></mrow></mpadded></msub><mo>↖</mo></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>j</mi> <mn>1</mn></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>k</mi> <mn>0</mn></msub></mrow></mpadded></msub><mo>↖</mo></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>j</mi> <mn>0</mn></msub></mrow></mpadded></msub><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mi>ℰ</mi> <mrow><mn>3</mn><mo>,</mo><mo>•</mo></mrow></msup></mtd> <mtd></mtd> <mtd><msup><mi>ℰ</mi> <mrow><mn>2</mn><mo>,</mo><mo>•</mo></mrow></msup></mtd> <mtd></mtd> <mtd><msup><mi>ℰ</mi> <mrow><mn>1</mn><mo>,</mo><mo>•</mo></mrow></msup></mtd> <mtd></mtd> <mtd><msup><mi>ℰ</mi> <mrow><mn>0</mn><mo>,</mo><mo>•</mo></mrow></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \cdots &\stackrel{}{\longrightarrow}& \mathcal{D}^{3,\bullet} &\stackrel{i_2}{\longrightarrow}& \mathcal{D}^{2,\bullet} &\stackrel{i_1}{\longrightarrow} & \mathcal{D}^{1,\bullet} &\stackrel{i_0}{\longrightarrow}& \mathcal{D}^{0,\bullet} \\ && \downarrow^{\mathrlap{}} &{}_{\mathllap{k_2}}\nwarrow & {}^{\mathllap{j_2}}\downarrow &{}_{\mathllap{k_1}}\nwarrow & {}^{\mathllap{j_1}}\downarrow &{}_{\mathllap{k_0}}\nwarrow & {}_{\mathllap{j_0}}\downarrow \\ && \mathcal{E}^{3,\bullet} && \mathcal{E}^{2,\bullet} && \mathcal{E}^{1,\bullet} && \mathcal{E}^{0,\bullet} } </annotation></semantics></math></div> <p>such that each triangle is a rolled-up <a class="existingWikiWord" href="/nlab/show/long+exact+sequence">long exact sequence</a> of <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</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>⋯</mi><mo>→</mo><msup><mi>𝒟</mi> <mrow><mi>s</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msup><mover><mo>⟶</mo><mrow><msub><mi>i</mi> <mi>s</mi></msub></mrow></mover><msup><mi>𝒟</mi> <mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msup><mover><mo>⟶</mo><mrow><msub><mi>j</mi> <mi>s</mi></msub></mrow></mover><msup><mi>ℰ</mi> <mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msup><mover><mo>⟶</mo><mrow><msub><mi>k</mi> <mi>s</mi></msub></mrow></mover><msup><mi>𝒟</mi> <mrow><mi>s</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>t</mi></mrow></msup><mo>→</mo><mi>⋯</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \cdots \to \mathcal{D}^{s+1,t+1} \stackrel{i_s}{\longrightarrow} \mathcal{D}^{s,t} \stackrel{j_s}{\longrightarrow} \mathcal{E}^{s,t} \stackrel{k_s}{\longrightarrow} \mathcal{D}^{s+1,t} \to \cdots \,. </annotation></semantics></math></div></div> <p>The collection of this “un-rolled” data into a single diagram of <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a> is called the corresponding <em><a class="existingWikiWord" href="/nlab/show/exact+couple">exact couple</a></em>.</p> <div class="num_defn" id="ExactCouple"> <h6 id="definition_7">Definition</h6> <p>An <em><a class="existingWikiWord" href="/nlab/show/exact+couple">exact couple</a></em> is a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> (non-commuting) of <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>𝒟</mi></mtd> <mtd><mover><mo>⟶</mo><mi>i</mi></mover></mtd> <mtd><mi>𝒟</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>k</mi></mpadded></msub><mo>↖</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>j</mi></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>ℰ</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{D} &\stackrel{i}{\longrightarrow}& \mathcal{D} \\ & {}_{\mathllap{k}}\nwarrow & \downarrow^{\mathrlap{j}} \\ && \mathcal{E} } \,, </annotation></semantics></math></div> <p>such that this is <a class="existingWikiWord" href="/nlab/show/exact+sequence">exact sequence</a> exact in each position, hence such that the <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> of every <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> is the <a class="existingWikiWord" href="/nlab/show/image">image</a> of the preceding one.</p> </div> <p>The concept of exact couple so far just collects the sequences of long exact sequences given by a filtration. Next we turn to extracting information from this sequence of sequences.</p> <div class="num_remark" id="Observingd1"> <h6 id="remark_5">Remark</h6> <p>The sequence of long exact sequences in remark <a class="maruku-ref" href="#UnrolledExactCoupleOfAFiltrationOnASpectrum"></a> is inter-locking, in that every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mrow><mi>t</mi><mo>−</mo><mi>s</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>s</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_{t-s}(X_s)</annotation></semantics></math> appears <em>twice</em>:</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><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↗</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mi>π</mi> <mrow><mi>t</mi><mo>−</mo><mi>s</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msubsup><mi>δ</mi> <mrow><mi>t</mi><mo>−</mo><mi>s</mi></mrow> <mi>s</mi></msubsup></mrow></mpadded></msup><mo>↗</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><msub><mi>π</mi> <mrow><mi>t</mi><mo>−</mo><mi>s</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>cofib</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>↗</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>π</mi> <mrow><mi>t</mi><mo>−</mo><mi>s</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>s</mi></msub><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><munder><mo>⟶</mo><mrow><mi>def</mi><mo>:</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msubsup><mi>d</mi> <mn>1</mn> <mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msubsup></mrow></munder></mtd> <mtd></mtd> <mtd><msub><mi>π</mi> <mrow><mi>t</mi><mo>−</mo><mi>s</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>def</mi><mo>:</mo><mspace width="thickmathspace"></mspace><msubsup><mi>d</mi> <mn>1</mn> <mrow><mi>s</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>t</mi></mrow></msubsup></mrow></mover></mtd> <mtd></mtd> <mtd><msub><mi>π</mi> <mrow><mi>t</mi><mo>−</mo><mi>s</mi><mo>−</mo><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mrow><mi>s</mi><mo>+</mo><mn>2</mn></mrow></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msubsup><mi>δ</mi> <mrow><mi>t</mi><mo>−</mo><mi>s</mi><mo>−</mo><mn>1</mn></mrow> <mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msubsup></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mrow><msub><mi>π</mi> <mrow><mi>t</mi><mo>−</mo><mi>s</mi><mo>−</mo><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><mi>cofib</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mrow><mi>s</mi><mo>+</mo><mn>2</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mi>π</mi> <mrow><mi>t</mi><mo>−</mo><mi>s</mi><mo>−</mo><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mrow><mi>s</mi><mo>+</mo><mn>2</mn></mrow></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && & \searrow && \nearrow \\ && && \pi_{t-s-1}(X_{s+1}) \\ && & {}^{\mathllap{\delta_{t-s}^s}}\nearrow && \searrow^{\mathrlap{\pi_{t-s-1}(cofib(f_{s+1}))}} && && && \nearrow \\ && \pi_{t-s}(A_s) && \underset{def: \;\;d_1^{s,t}}{\longrightarrow} && \pi_{t-s-1}(A_{s+1}) && \stackrel{def: \; d_1^{s+1,t}}{\longrightarrow} && \pi_{t-s-2}(A_{s+2}) \\ & \nearrow && && && {}_{\mathllap{\delta_{t-s-1}^{s+1}}}\searrow && \nearrow_{\mathrlap{\pi_{t-s-2}(cofib(f_{s+2}))}} \\ && && && && \pi_{t-s-2}(X_{s+2}) \\ && && && & \nearrow && \searrow } </annotation></semantics></math></div> <p>This gives rise to the horizontal composites <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>d</mi> <mn>1</mn> <mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">d_1^{s,t}</annotation></semantics></math>, as show above, and by the fact that the diagonal sequences are long exact, these are differentials: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>d</mi> <mn>1</mn> <mn>2</mn></msubsup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">d_1^2 = 0</annotation></semantics></math>, hence give a <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</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>⋯</mi></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd></mtd> <mtd><msub><mi>π</mi> <mrow><mi>t</mi><mo>−</mo><mi>s</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>s</mi></msub><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mrow><msubsup><mi>d</mi> <mn>1</mn> <mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msubsup></mrow></mover></mtd> <mtd></mtd> <mtd><msub><mi>π</mi> <mrow><mi>t</mi><mo>−</mo><mi>s</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mrow><msubsup><mi>d</mi> <mn>1</mn> <mrow><mi>s</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>t</mi></mrow></msubsup></mrow></mover></mtd> <mtd></mtd> <mtd><msub><mi>π</mi> <mrow><mi>t</mi><mo>−</mo><mi>s</mi><mo>−</mo><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mrow><mi>s</mi><mo>+</mo><mn>2</mn></mrow></msub><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>⋯</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \cdots & \stackrel{}{\longrightarrow} && \pi_{t-s}(A_s) && \overset{d_1^{s,t}}{\longrightarrow} && \pi_{t-s-1}(A_{s+1}) && \stackrel{d_1^{s+1,t}}{\longrightarrow} && \pi_{t-s-2}(A_{s+2}) &&\longrightarrow & \cdots } \,. </annotation></semantics></math></div> <p>We read off from the interlocking long exact sequences what these differentials <em>mean</em>: an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><msub><mi>π</mi> <mrow><mi>t</mi><mo>−</mo><mi>s</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>s</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c \in \pi_{t-s}(A_s)</annotation></semantics></math> lifts to an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>c</mi><mo stretchy="false">^</mo></mover><mo>∈</mo><msub><mi>π</mi> <mrow><mi>t</mi><mo>−</mo><mi>s</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mrow><mi>s</mi><mo>+</mo><mn>2</mn></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\hat c \in \pi_{t-s-1}(X_{s+2})</annotation></semantics></math> precisely if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mn>1</mn></msub><mi>c</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">d_1 c = 0</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd><mover><mi>c</mi><mo stretchy="false">^</mo></mover><mo>∈</mo></mtd> <mtd><msub><mi>π</mi> <mrow><mi>t</mi><mo>−</mo><mi>s</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mrow><mi>s</mi><mo>+</mo><mn>2</mn></mrow></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><msub><mi>π</mi> <mrow><mi>t</mi><mo>−</mo><mi>s</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>f</mi> <mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mi>π</mi> <mrow><mi>t</mi><mo>−</mo><mi>s</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msubsup><mi>δ</mi> <mrow><mi>t</mi><mo>−</mo><mi>s</mi></mrow> <mi>s</mi></msubsup></mrow></mpadded></msup><mo>↗</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><msub><mi>π</mi> <mrow><mi>t</mi><mo>−</mo><mi>s</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>cofib</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mi>c</mi><mo>∈</mo></mtd> <mtd><msub><mi>π</mi> <mrow><mi>t</mi><mo>−</mo><mi>s</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>s</mi></msub><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><munder><mo>⟶</mo><mrow><msubsup><mi>d</mi> <mn>1</mn> <mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msubsup></mrow></munder></mtd> <mtd></mtd> <mtd><msub><mi>π</mi> <mrow><mi>t</mi><mo>−</mo><mi>s</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ &\hat c \in & \pi_{t-s-1}(X_{s+2}) \\ && & \searrow^{\mathrlap{\pi_{t-s-1}(f_{s+1})}} \\ && && \pi_{t-s-1}(X_{s+1}) \\ && & {}^{\mathllap{\delta_{t-s}^s}}\nearrow && \searrow^{\mathrlap{\pi_{t-s-1}(cofib(f_{s+1}))}} \\ & c \in & \pi_{t-s}(A_s) && \underset{d_1^{s,t}}{\longrightarrow} && \pi_{t-s-1}(A_{s+1}) } </annotation></semantics></math></div> <p>This means that the <a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a> of the complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>,</mo><msub><mi>d</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\pi_{\bullet}(A_\bullet), d_1)</annotation></semantics></math> produces elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mo>•</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_\bullet(X_\bullet)</annotation></semantics></math> and hence of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_\bullet(X)</annotation></semantics></math>.</p> <p>In order to organize this observation, notice that in terms of the exact couple of remark <a class="maruku-ref" href="#UnrolledExactCoupleOfAFiltrationOnASpectrum"></a>, the differential</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> <mn>1</mn> <mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msubsup><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msub><mi>π</mi> <mrow><mi>t</mi><mo>−</mo><mi>s</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>cofib</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∘</mo><msubsup><mi>δ</mi> <mrow><mi>t</mi><mo>−</mo><mi>s</mi></mrow> <mi>s</mi></msubsup></mrow><annotation encoding="application/x-tex"> d_1^{s,t} \;\coloneqq \; \pi_{t-s-1}(cofib(f_{s+1})) \circ \delta_{t-s}^s </annotation></semantics></math></div> <p>is a component of the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>≔</mo><mi>j</mi><mo>∘</mo><mi>k</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> d \coloneqq j \circ k \,. </annotation></semantics></math></div></div> <p>Some terminology:</p> <div class="num_defn" id="PageOfAnExactCouple"> <h6 id="definition_8">Definition</h6> <p>Given an exact couple, def. <a class="maruku-ref" href="#ExactCouple"></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><msup><mi>𝒟</mi> <mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msup></mtd> <mtd><mover><mo>⟶</mo><mi>i</mi></mover></mtd> <mtd><msup><mi>𝒟</mi> <mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>k</mi></mpadded></msub><mo>↖</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>j</mi></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mi>ℰ</mi> <mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{D}^{\bullet,\bullet} &\stackrel{i}{\longrightarrow}& \mathcal{D}^{\bullet,\bullet} \\ & {}_{\mathllap{k}}\nwarrow & \downarrow^{\mathrlap{j}} \\ && \mathcal{E}^{\bullet,\bullet} } </annotation></semantics></math></div> <p>its <em>page</em> is the <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a></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><msup><mi>E</mi> <mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msup><mo>,</mo><mi>d</mi><mo>≔</mo><mi>j</mi><mo>∘</mo><mi>k</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (E^{\bullet,\bullet}, d \coloneqq j \circ k) \,. </annotation></semantics></math></div></div> <div class="num_defn" id="DerivedExactCouple"> <h6 id="definition_9">Definition</h6> <p>Given an exact couple, def. <a class="maruku-ref" href="#ExactCouple"></a>, then the induced <em>derived exact couple</em> is the 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><mover><mi>𝒟</mi><mo>˜</mo></mover></mtd> <mtd><mover><mo>⟶</mo><mover><mi>i</mi><mo stretchy="false">˜</mo></mover></mover></mtd> <mtd><mover><mi>𝒟</mi><mo>˜</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mover><mi>k</mi><mo stretchy="false">˜</mo></mover></mpadded></msub><mo>↖</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mover><mi>j</mi><mo stretchy="false">˜</mo></mover></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mover><mi>ℰ</mi><mo>˜</mo></mover></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \widetilde {\mathcal{D}} &\stackrel{\tilde i}{\longrightarrow}& \widetilde {\mathcal{D}} \\ & {}_{\mathllap{\tilde k}}\nwarrow & \downarrow^{\mathrlap{\tilde j}} \\ && \widetilde{\mathcal{E}} } </annotation></semantics></math></div> <p>with</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>ℰ</mi><mo stretchy="false">˜</mo></mover><mo>≔</mo><mi>ker</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>im</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tilde{\mathcal{E}} \coloneqq ker(d)/im(d)</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>𝒟</mi><mo stretchy="false">˜</mo></mover><mo>≔</mo><mi>im</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tilde {\mathcal{D}} \coloneqq im(i)</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>i</mi><mo stretchy="false">˜</mo></mover><mo>≔</mo><mi>i</mi><msub><mo stretchy="false">|</mo> <mrow><mi>im</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\tilde i \coloneqq i|_{im(i)}</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>j</mi><mo stretchy="false">˜</mo></mover><mo>≔</mo><mi>j</mi><mo>∘</mo><mo stretchy="false">(</mo><mi>im</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\tilde j \coloneqq j \circ (im(i))^{-1}</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>k</mi><mo stretchy="false">˜</mo></mover><mo>≔</mo><mi>k</mi><msub><mo stretchy="false">|</mo> <mrow><mi>ker</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\tilde k \coloneqq k|_{ker(d)}</annotation></semantics></math>.</p> </li> </ol> </div> <div class="num_prop" id="DerivedExactCoupleIsExactCouple"> <h6 id="proposition_2">Proposition</h6> <p>A derived exact couple, def. <a class="maruku-ref" href="#DerivedExactCouple"></a>, is again an exact couple, def. <a class="maruku-ref" href="#ExactCouple"></a>.</p> </div> <div class="num_defn"> <h6 id="definition_10">Definition</h6> <p>Given an exact couple, def. <a class="maruku-ref" href="#ExactCouple"></a>, then the induced <a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a>, def. <a class="maruku-ref" href="#SpectralSequence"></a>, is the sequence of pages, def. <a class="maruku-ref" href="#PageOfAnExactCouple"></a>, of the induced sequence of derived exact couples, def. <a class="maruku-ref" href="#DerivedExactCouple"></a>, prop. <a class="maruku-ref" href="#DerivedExactCoupleIsExactCouple"></a>.</p> </div> <div class="num_example" id="AdamsTypeSpectralSequenceOfATower"> <h6 id="example">Example</h6> <p>Consider a <a class="existingWikiWord" href="/nlab/show/filtered+spectrum">filtered spectrum</a>, def. <a class="maruku-ref" href="#FilteredSpectrum"></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>⋯</mi></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><msub><mi>X</mi> <mn>3</mn></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>f</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msub><mi>X</mi> <mn>2</mn></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>f</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msub><mi>X</mi> <mn>1</mn></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>A</mi> <mn>3</mn></msub></mtd> <mtd></mtd> <mtd><msub><mi>A</mi> <mn>2</mn></msub></mtd> <mtd></mtd> <mtd><msub><mi>A</mi> <mn>1</mn></msub></mtd> <mtd></mtd> <mtd><msub><mi>A</mi> <mn>0</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \cdots &\stackrel{}{\longrightarrow}& X_3 &\stackrel{f_2}{\longrightarrow}& X_2 &\stackrel{f_2}{\longrightarrow} & X_1 &\stackrel{f_1}{\longrightarrow}& X \\ && \downarrow && \downarrow && \downarrow && \downarrow \\ && A_3 && A_2 && A_1 && A_0 } </annotation></semantics></math></div> <p>and its induced <a class="existingWikiWord" href="/nlab/show/exact+couple">exact couple</a> of <a class="existingWikiWord" href="/nlab/show/stable+homotopy+groups">stable homotopy groups</a>, from remark <a class="maruku-ref" href="#UnrolledExactCoupleOfAFiltrationOnASpectrum"></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>𝒟</mi></mtd> <mtd><mover><mo>⟶</mo><mi>i</mi></mover></mtd> <mtd><mi>𝒟</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>k</mi></mpadded></msub><mo>↖</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>j</mi></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>ℰ</mi></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><mi>𝒟</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>𝒟</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mpadded></msub><mo>↖</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>ℰ</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{D} &\stackrel{i}{\longrightarrow}& \mathcal{D} \\ &{}_{\mathllap{k}}\nwarrow& \downarrow^{\mathrlap{j}} \\ && \mathcal{E} } \;\;\;\;\;\,\;\;\;\;\;\; \array{ \mathcal{D} &\stackrel{(-1,-1)}{\longrightarrow}& \mathcal{D} \\ &{}_{\mathllap{(1,0)}}\nwarrow& \downarrow^{\mathrlap{(0,0)}} \\ && \mathcal{E} } </annotation></semantics></math></div> <p>with bigrading as shown on the right.</p> <div style="float:right;margin:0 10px 10px 0;"> <img src="http://ncatlab.org/nlab/files/adamstypedifferentials.jpg" width="360" /> </div> <p>As we pass to derived exact couples, by def. <a class="maruku-ref" href="#DerivedExactCouple"></a>, the bidegree of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> is preserved, but that of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math> increases by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1,1)</annotation></semantics></math> in each step, since</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>deg</mi><mo stretchy="false">(</mo><mover><mi>j</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mi>deg</mi><mo stretchy="false">(</mo><mi>j</mi><mo>∘</mo><mi>im</mi><mo stretchy="false">(</mo><mi>i</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>=</mo><mi>deg</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> deg(\tilde j) = deg( j \circ im(i)^{-1}) = deg(j) + (1,1) \,. </annotation></semantics></math></div> <p>Therefore the induced <a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a> has differentials of the form</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> <mi>r</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msubsup><mi>ℰ</mi> <mi>r</mi> <mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msubsup><mo>⟶</mo><msubsup><mi>ℰ</mi> <mi>r</mi> <mrow><mi>s</mi><mo>+</mo><mi>r</mi><mo>,</mo><mi>t</mi><mo>+</mo><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msubsup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> d_r \;\colon\; \mathcal{E}_r^{s,t} \longrightarrow \mathcal{E}_r^{s+r, t+r-1} \,. </annotation></semantics></math></div></div> <h2 id="references">References</h2> <p>The original article is</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/W.+S.+Massey">W. S. Massey</a>, <em>Exact Couples in Algebraic Topology (Parts I and II)</em>, Annals of Mathematics, Second Series, Vol. 56, No. 2 (Sep., 1952), pp. 363-396 (<a href="http://www.maths.ed.ac.uk/~aar/papers/massey6.pdf">pdf</a>)</li> </ul> <p>also</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Beno+Eckmann">Beno Eckmann</a>, <a class="existingWikiWord" href="/nlab/show/Peter+Hilton">Peter Hilton</a>, <em>Exact couples in an abelian category</em>, Journal of Algebra Volume 3, Issue 1, January 1966, Pages 38-87 (<a href="http://www.sciencedirect.com/science/article/pii/0021869366900196">jorunal</a>)</li> </ul> <p>A class of examples leading to what later came to be known as the <a class="existingWikiWord" href="/nlab/show/Atiyah-Hirzebruch+spectral+sequence">Atiyah-Hirzebruch spectral sequence</a> is discussed in section XV.7 of</p> <ul> <li id="CartanEilenberg56"><a class="existingWikiWord" href="/nlab/show/Henri+Cartan">Henri Cartan</a>, <a class="existingWikiWord" href="/nlab/show/Samuel+Eilenberg">Samuel Eilenberg</a>, <em>Homological algebra</em>, Princeton Univ. Press (1956)</li> </ul> <p>Textbook accounts include</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Frank+Adams">Frank Adams</a>, part III, section 7 of <em><a class="existingWikiWord" href="/nlab/show/Stable+homotopy+and+generalised+homology">Stable homotopy and generalised homology</a></em>, 1974</p> </li> <li id="Kochmann96"> <p><a class="existingWikiWord" href="/nlab/show/Stanley+Kochmann">Stanley Kochmann</a>, section 2.2 of <em><a class="existingWikiWord" href="/nlab/show/Bordism%2C+Stable+Homotopy+and+Adams+Spectral+Sequences">Bordism, Stable Homotopy and Adams Spectral Sequences</a></em>, AMS 1996</p> </li> <li id="McCleary01"> <p><a class="existingWikiWord" href="/nlab/show/John+McCleary">John McCleary</a>, section 2.2 (from p. 37 on) in <em><a href="https://pages.vassar.edu/mccleary/books/users-guide-to-spectral-sequences/">A user’s guide to spectral sequences</a></em>, Cambridge University Press, 2001</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Charles+Weibel">Charles Weibel</a>, section 5.9 <em><a class="existingWikiWord" href="/nlab/show/An+Introduction+to+Homological+Algebra">An Introduction to Homological Algebra</a></em></p> </li> </ul> <p>Another review with an eye towards application to the <a class="existingWikiWord" href="/nlab/show/Adams+spectral+sequence">Adams spectral sequence</a> is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Doug+Ravenel">Doug Ravenel</a>, chapter 2, section 1 of <em><a class="existingWikiWord" href="/nlab/show/Complex+cobordism+and+stable+homotopy+groups+of+spheres">Complex cobordism and stable homotopy groups of spheres</a></em></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 13, 2019 at 01:49:07. 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