CINXE.COM

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> spectral sequence in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="index,follow" /> <meta name="viewport" content="width=device-width, initial-scale=1" /> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/mathematics.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/syntax.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/nlab.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link rel="stylesheet" type="text/css" href="https://cdn.jsdelivr.net/gh/dreampulse/computer-modern-web-font@master/fonts.css"/> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } a:visited.existingWikiWord { color: #164416; } </style> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script src="/javascripts/page_helper.js?1660229990" type="text/javascript"></script> <script src="/javascripts/thm_numbering.js?1660229990" type="text/javascript"></script> <script type="text/x-mathjax-config"> <![CDATA[//><!]]> </script> <script type="text/javascript"> <![CDATA[//><!]]> </script> <link href="https://ncatlab.org/nlab/atom_with_headlines" rel="alternate" title="Atom with headlines" type="application/atom+xml" /> <link href="https://ncatlab.org/nlab/atom_with_content" rel="alternate" title="Atom with full content" type="application/atom+xml" /> <script type="text/javascript"> document.observe("dom:loaded", function() { generateThmNumbers(); }); </script> </head> <body> <div id="Container"> <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 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> spectral sequence </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/2161/#Item_34" 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> <h4 id="algebraic_topology">Algebraic topology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></strong> – application of <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a> and <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a> to the study of (<a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a>) <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy</a></p> <p><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> <p><a class="existingWikiWord" href="/nlab/show/homology">homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a></p> </li> </ul> </div></div> <h4 id="stable_homotopy_theory">Stable Homotopy theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a>, <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></li> </ul> <p><em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Stable+Homotopy+Theory">Introduction</a></em></p> <h1 id="ingredients">Ingredients</h1> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></li> </ul> <h1 id="contents">Contents</h1> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/suspension+object">suspension object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/looping+and+delooping">looping and delooping</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/stabilization">stabilization</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/spectrum+object">spectrum object</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+derivator">stable derivator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category+of+spectra">stable (∞,1)-category of spectra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+homotopy+category">stable homotopy category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+smash+product+of+spectra">symmetric smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Spanier-Whitehead+duality">Spanier-Whitehead duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/stable+homotopy+theory+-+contents">Edit this sidebar</a> </p> </div></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='#spectral_sequence'>Spectral sequence</a></li> <li><a href='#ConvergenceOfSpectralSequences'>Convergence</a></li> <li><a href='#Boundedness'>Boundedness</a></li> <li><a href='#ExtensionProblem'>Extension problem</a></li> </ul> <li><a href='#Examples'>Examples</a></li> <ul> <li><a href='#SpectralSequenceOfFilteredComplex'>Spectral sequence of a filtered complex</a></li> <li><a href='#SpectralSequenceOfADoubleComplex'>Spectral sequence of a double complex</a></li> <li><a href='#HyperDerivedFunctors'>Spectral sequences for hyper-derived functors</a></li> <li><a href='#GrothendieckSpectralSequence'>Grothendieck spectral sequence</a></li> <li><a href='#SpecialGrothendieckSpectralSequence'>Special Grothendieck spectral sequences</a></li> <ul> <li><a href='#LeraySpectralSequence'>Leray spectral sequence</a></li> <li><a href='#BaseChangeSpectralSequence'>Base change spectral sequence for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Tor</mi></mrow><annotation encoding="application/x-tex">Tor</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ext</mi></mrow><annotation encoding="application/x-tex">Ext</annotation></semantics></math></a></li> <li><a href='#HochschildSerreSpectralSequence'>Hochschild-Serre spectral sequence</a></li> </ul> <li><a href='#exact_couples'>Exact couples</a></li> <li><a href='#ListOfExamples'>List of examples</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#basic_lemmas'>Basic lemmas</a></li> <li><a href='#FirstQuadrant'>First quadrant spectral sequence</a></li> <li><a href='#PropertiesCupProductStructure'>Cup product structure</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general_exposition'>General exposition</a></li> <li><a href='#abelianstable_theory'>Abelian/stable theory</a></li> <li><a href='#ReferencesNonabelian'>Nonabelian / unstable theory</a></li> <li><a href='#history'>History</a></li> </ul> </ul> </div> <h2 id="Idea">Idea</h2> <p>The notion of <em>spectral sequence</em> is an <a class="existingWikiWord" href="/nlab/show/algorithm">algorithm</a> or computational tool in <a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a> and more generally in <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> which allows to compute <a class="existingWikiWord" href="/nlab/show/chain+homology">chain homology groups</a>/<a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> of <em>bi</em>-<a class="existingWikiWord" href="/nlab/show/graded+objects">graded objects</a> from the homology/homotopy of the two graded components.</p> <p>Notably there is a spectral sequence for computing the homology of the <a class="existingWikiWord" href="/nlab/show/total+complex">total complex</a> of a <a class="existingWikiWord" href="/nlab/show/double+complex">double complex</a> from the homology of its row and column complexes separately. This in turn allows to compute <a class="existingWikiWord" href="/nlab/show/derived+functors">derived functors</a> of composite functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>∘</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">G\circ F</annotation></semantics></math> from the double complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mo>•</mo></msup><mi>G</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mo>•</mo></msup><mi>F</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}^\bullet G (\mathbb{R}^\bullet F(-))</annotation></semantics></math> obtained by non-totally deriving the two functors separately (called the <a href="#GrothendieckSpectralSequence">Grothendieck spectral sequence</a>). By choosing various functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> and <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> here this gives rise to various important classes of examples of spectral sequences, see <a href="#SpecialGrothendieckSpectralSequence">below</a>.</p> <p>More concretely, a homology spectral sequence is a sequence of graded chain complexes that provides the higher order corrections to the <em>naïve</em> idea of computing the homology of the <a class="existingWikiWord" href="/nlab/show/total+complex">total complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Tot</mi><mo stretchy="false">(</mo><mi>V</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">Tot(V)_\bullet</annotation></semantics></math> of a <a class="existingWikiWord" href="/nlab/show/double+complex">double complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msub></mrow><annotation encoding="application/x-tex">V_{\bullet, \bullet}</annotation></semantics></math>: by first computing those of the vertical differential, then those of the horizontal differential induced on these vertical homology groups (or the other way around). This simple idea in general does not produce the correct homology groups of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Tot</mi><mo stretchy="false">(</mo><mi>V</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">Tot(V)_\bullet</annotation></semantics></math>, but it does produce a “first-order approximation” to them, in a useful sense. The spectral sequence is the sequence of higher-order corrections that make this naive idea actually work.</p> <p>Being, therefore, an iterative perturbative approximation scheme of bigraded differential objects, fully-fledged spectral sequences can look a bit intricate. However, a standard experience in mathematical practice is that for most problems of practical interest the relevant spectral sequence “perturbation series” yields the exact result already at the second stage. This reduces the computational complexity immensely and makes spectral sequences a wide-spread useful computational tool.</p> <p>Despite their name, there seemed to be nothing specifically “spectral” about spectral sequences, for any of the technical meanings of the word <a class="existingWikiWord" href="/nlab/show/spectrum+-+disambiguation">spectrum</a>. Together with the concept, this term was introduced by <a class="existingWikiWord" href="/nlab/show/Jean+Leray">Jean Leray</a> and has long become standard, but was never really motivated (see p. 5 of <a href="#Chow">Chow</a>). But then, by lucky coincidence it turns out in the refined context of <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a> theory/<a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a> that spectral sequences frequently arise by considering the <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> of <em><a class="existingWikiWord" href="/nlab/show/sequential+diagram">sequences</a> of <a class="existingWikiWord" href="/nlab/show/spectra">spectra</a></em>. This is discussed at <em><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+filtered+stable+homotopy+type">spectral sequence of a filtered stable homotopy type</a></em>.</p> <p>While therefore spectral sequences are a notion considered in the context of <a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a> and more generally in <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a>, there is also an “unstable” or <a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian</a> variant of the notion in plain <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, called <em><a class="existingWikiWord" href="/nlab/show/homotopy+spectral+sequence">homotopy spectral sequence</a></em>.</p> <h2 id="definition">Definition</h2> <p>We give the general definition of a (co)homology spectral sequence. For motivation see the example <em><a href="#SpectralSequenceOfFilteredComplex">Spectral sequence of a filtered complex</a></em> below.</p> <p>Throughout, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>.</p> <h3 id="spectral_sequence">Spectral sequence</h3> <div class="num_defn" id="CohomologySpectralSequence"> <h6 id="definition_2">Definition</h6> <p>A <strong>cohomology spectral sequence</strong> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> is</p> <ul> <li> <p>a family <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 stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E^{p,q}_r)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/objects">objects</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math>, for all <a class="existingWikiWord" href="/nlab/show/integers">integers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi>r</mi></mrow><annotation encoding="application/x-tex">p,q,r</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">r\geq 1</annotation></semantics></math></p> <p>(for a fixed <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> these are said to form the <strong><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>-th page</strong> of the spectral sequence)</p> </li> <li> <p>for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi>r</mi></mrow><annotation encoding="application/x-tex">p,q,r</annotation></semantics></math> as above a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> (called the <strong>differential</strong>)</p> <div class="maruku-equation" id="eq:FormOfDifferentialInCohomologySpectralSequence"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>d</mi> <mi>r</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msubsup><mi>E</mi> <mi>r</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup><mo>⟶</mo><msubsup><mi>E</mi> <mi>r</mi> <mrow><mi>p</mi><mo>+</mo><mi>r</mi><mo>,</mo><mi>q</mi><mo>−</mo><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex"> d^{p,q}_r \;\colon\; E^{p,q}_r \longrightarrow E^{p+r,q-r+1}_r </annotation></semantics></math></div> <p>satisfying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>d</mi> <mi>r</mi> <mn>2</mn></msubsup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">d_r^2 = 0</annotation></semantics></math> (more precisely, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>d</mi> <mi>r</mi> <mrow><mi>p</mi><mo>+</mo><mi>r</mi><mo>,</mo><mi>q</mi><mo>−</mo><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo>∘</mo><msubsup><mi>d</mi> <mi>r</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">d_r^{p+r,q-r+1}\circ d_r^{p,q} = 0</annotation></semantics></math>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>α</mi> <mi>r</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup><mo>:</mo><msup><mi>H</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msup><mo stretchy="false">(</mo><msub><mi>E</mi> <mi>r</mi></msub><mo stretchy="false">)</mo><mo>→</mo><msubsup><mi>E</mi> <mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">\alpha_r^{p,q}: H^{p,q}(E_r)\to E^{p,q}_{r+1}</annotation></semantics></math> where</p> <p>the <a class="existingWikiWord" href="/nlab/show/chain+cohomology">chain cohomology</a> is given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msup><mo stretchy="false">(</mo><msub><mi>E</mi> <mi>r</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">ker</mi><msubsup><mi>d</mi> <mi>r</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup><mo stretchy="false">/</mo><mi mathvariant="normal">im</mi><msubsup><mi>d</mi> <mi>r</mi> <mrow><mi>p</mi><mo>−</mo><mi>r</mi><mo>,</mo><mi>q</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"> H^{p,q}(E_r) = \mathrm{ker} d^{p,q}_r/ \mathrm{im} d^{p-r,q+r-1}_r \,. </annotation></semantics></math></div></li> </ul> </div> <p>Analogously, a <strong>homology spectral sequence</strong> is collection of objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>E</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow> <mi>r</mi></msubsup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E_{p,q}^r)</annotation></semantics></math> with the differential <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> of degree <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><mi>r</mi><mo>,</mo><mi>r</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-r,r-1)</annotation></semantics></math>.</p> <h3 id="ConvergenceOfSpectralSequences">Convergence</h3> <div class="num_defn" id="LimitTerm"> <h6 id="definition_3">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msubsup><mi>E</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow> <mi>r</mi></msubsup><msub><mo stretchy="false">}</mo> <mrow><mi>r</mi><mo>,</mo><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{E^r_{p,q}\}_{r,p,q}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a> such that for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow><annotation encoding="application/x-tex">p,q</annotation></semantics></math> there is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">r(p,q)</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><mi>r</mi><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">r \geq r(p,q)</annotation></semantics></math> we have</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> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow> <mrow><mi>r</mi><mo>≥</mo><mi>r</mi><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>≃</mo><msubsup><mi>E</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow> <mrow><mi>r</mi><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow></msubsup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> E^{r \geq r(p,q)}_{p,q} \simeq E^{r(p,q)}_{p,q} \,. </annotation></semantics></math></div> <p>Then one says equivalently that</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/bigraded+object">bigraded object</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mn>∞</mn></msup><mo>≔</mo><mo stretchy="false">{</mo><msubsup><mi>E</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow> <mn>∞</mn></msubsup><msub><mo stretchy="false">}</mo> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msub><mo>≔</mo><mo stretchy="false">{</mo><msubsup><mi>E</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow> <mrow><mi>r</mi><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow></msubsup><msub><mo stretchy="false">}</mo> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> E^\infty \coloneqq \{E^\infty_{p,q}\}_{p,q} \coloneqq \{ E^{r(p,q)}_{p,q} \}_{p,q} </annotation></semantics></math></div> <p>is the <strong>limit term</strong> of the spectral sequence;</p> </li> <li> <p>the spectral sequence <strong>abuts</strong> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">E^\infty</annotation></semantics></math>.</p> </li> </ol> </div> <div class="num_example" id="Degeneration"> <h6 id="example">Example</h6> <p>If for a spectral sequence there is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>r</mi> <mi>s</mi></msub></mrow><annotation encoding="application/x-tex">r_s</annotation></semantics></math> such that all <a class="existingWikiWord" href="/nlab/show/differentials">differentials</a> on pages after <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>r</mi> <mi>s</mi></msub></mrow><annotation encoding="application/x-tex">r_s</annotation></semantics></math> vanish, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∂</mo> <mrow><mi>r</mi><mo>≥</mo><msub><mi>r</mi> <mi>s</mi></msub></mrow></msup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\partial^{r \geq r_s} = 0</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>E</mi> <mrow><msub><mi>r</mi> <mi>s</mi></msub></mrow></msup><msub><mo stretchy="false">}</mo> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{E^{r_s}\}_{p,q}</annotation></semantics></math> is a limit term for the spectral sequence. One says in this cases that the spectral sequence <strong>degenerates</strong> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>r</mi> <mi>s</mi></msub></mrow><annotation encoding="application/x-tex">r_s</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>By the defining relation</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> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow> <mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo>≃</mo><mi>ker</mi><mo stretchy="false">(</mo><msubsup><mo>∂</mo> <mrow><mi>p</mi><mo>−</mo><mi>r</mi><mo>,</mo><mi>q</mi><mo>+</mo><mi>r</mi><mo>−</mo><mn>1</mn></mrow> <mi>r</mi></msubsup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>im</mi><mo stretchy="false">(</mo><msubsup><mo>∂</mo> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow> <mi>r</mi></msubsup><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>E</mi> <mi>pq</mi> <mi>r</mi></msubsup></mrow><annotation encoding="application/x-tex"> E^{r+1}_{p,q} \simeq ker(\partial^r_{p-r,q+r-1})/im(\partial^r_{p,q}) = E^r_{pq} </annotation></semantics></math></div> <p>the spectral sequence becomes constant in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>r</mi> <mi>s</mi></msub></mrow><annotation encoding="application/x-tex">r_s</annotation></semantics></math> on if all the differentials vanish, so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ker</mi><mo stretchy="false">(</mo><msubsup><mo>∂</mo> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow> <mi>r</mi></msubsup><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>E</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow> <mi>r</mi></msubsup></mrow><annotation encoding="application/x-tex">ker(\partial^r_{p,q}) = E^r_{p,q}</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow><annotation encoding="application/x-tex">p,q</annotation></semantics></math>.</p> </div> <div class="num_example" id="Collaps"> <h6 id="example_2">Example</h6> <p>If for a <a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msubsup><mi>E</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow> <mi>r</mi></msubsup><msub><mo stretchy="false">}</mo> <mrow><mi>r</mi><mo>,</mo><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{E^r_{p,q}\}_{r,p,q}</annotation></semantics></math> there is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>r</mi> <mi>s</mi></msub><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">r_s \geq 2</annotation></semantics></math> such that the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>r</mi> <mi>s</mi></msub></mrow><annotation encoding="application/x-tex">r_s</annotation></semantics></math>th page is concentrated in a single row or a single column, then the spectral sequence degenerates on this pages, example <a class="maruku-ref" href="#Degeneration"></a>, hence this page is a limit term, def. <a class="maruku-ref" href="#LimitTerm"></a>. One says in this case that the spectral sequence <strong>collapses</strong> on this page.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>For <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> the <a class="existingWikiWord" href="/nlab/show/differentials">differentials</a> of the spectral sequence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mo>∂</mo> <mi>r</mi></msup><mo lspace="verythinmathspace">:</mo><msubsup><mi>E</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow> <mi>r</mi></msubsup><mo>→</mo><msubsup><mi>E</mi> <mrow><mi>p</mi><mo>−</mo><mi>r</mi><mo>,</mo><mi>q</mi><mo>+</mo><mi>r</mi><mo>−</mo><mn>1</mn></mrow> <mi>r</mi></msubsup></mrow><annotation encoding="application/x-tex"> \partial^r \colon E^r_{p,q} \to E^r_{p-r, q+r-1} </annotation></semantics></math></div> <p>have <a class="existingWikiWord" href="/nlab/show/domain">domain</a> and <a class="existingWikiWord" href="/nlab/show/codomain">codomain</a> necessarily in different rows an columns (while for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">r = 1</annotation></semantics></math> both are in the same row and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">r = 0</annotation></semantics></math> both coincide). Therefore if all but one row or column vanish, then all these differentials vanish.</p> </div> <div class="num_defn" id="Convergence"> <h6 id="definition_4">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msubsup><mi>E</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow> <mi>r</mi></msubsup><msub><mo stretchy="false">}</mo> <mrow><mi>r</mi><mo>,</mo><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{E^r_{p,q}\}_{r,p,q}</annotation></semantics></math> <strong>converges weakly</strong> to a <a class="existingWikiWord" href="/nlab/show/graded+object">graded object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">H_\bullet</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/exhaustive+filtering">exhaustive filtering</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mo>•</mo></msub><msub><mi>H</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">F_\bullet H_\bullet</annotation></semantics></math>, traditionally 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> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow> <mi>r</mi></msubsup><mo>⇒</mo><msub><mi>H</mi> <mo>•</mo></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> E^r_{p,q} \Rightarrow H_\bullet \,, </annotation></semantics></math></div> <p>if the <a class="existingWikiWord" href="/nlab/show/associated+graded">associated graded</a> complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>G</mi> <mi>p</mi></msub><msub><mi>H</mi> <mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></msub><msub><mo stretchy="false">}</mo> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msub><mo>≔</mo><mo stretchy="false">{</mo><msub><mi>F</mi> <mi>p</mi></msub><msub><mi>H</mi> <mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></msub><mo stretchy="false">/</mo><msub><mi>F</mi> <mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>H</mi> <mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{G_p H_{p+q}\}_{p,q} \coloneqq \{F_p H_{p+q} / F_{p-1} H_{p+q}\}</annotation></semantics></math> of <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> is the limit term of <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>, def. <a class="maruku-ref" href="#LimitTerm"></a>:</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> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow> <mn>∞</mn></msubsup><mo>≃</mo><msub><mi>G</mi> <mi>p</mi></msub><msub><mi>H</mi> <mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></msub><mspace width="thickmathspace"></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><msub><mo>∀</mo> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> E^\infty_{p,q} \simeq G_p H_{p+q} \;\;\;\;\;\;\; \forall_{p,q} \,. </annotation></semantics></math></div> <p>Furthermore one says that <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></p> <ul> <li> <p><strong>converges</strong> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">F_\bullet</annotation></semantics></math> is also a <a class="existingWikiWord" href="/nlab/show/Hausdorff+filtration">Hausdorff filtration</a>;</p> </li> <li> <p><strong>converges strongly</strong> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">F_\bullet</annotation></semantics></math> is also a <a class="existingWikiWord" href="/nlab/show/complete+Hausdorff+filtration">complete Hausdorff filtration</a>.</p> </li> </ul> </div> <p>(<a href="#Boardman99">Boardman 99, def. 2.5</a>) See also <em><a class="existingWikiWord" href="/nlab/show/conditional+convergence+of+spectral+sequences">conditional convergence of spectral sequences</a></em>.</p> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>In practice, spectral sequences are often referred to via their first interesting page, usually the first or the second. Then one often uses notation such as</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> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow> <mn>1</mn></msubsup><mspace width="thickmathspace"></mspace><mo>⇒</mo><mspace width="thickmathspace"></mspace><msub><mi>H</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex"> E^1_{p,q} \;\Rightarrow\; H_\bullet </annotation></semantics></math></div> <p>or</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> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow> <mn>2</mn></msubsup><mspace width="thickmathspace"></mspace><mo>⇒</mo><mspace width="thickmathspace"></mspace><msub><mi>H</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex"> E^2_{p,q} \;\Rightarrow\; H_\bullet </annotation></semantics></math></div> <p>to be read as “There is a spectral sequence whose first/second page is as shown on the left and which converges (weakly, strongly, or conditionally) to a filtered object as shown on the right.”</p> </div> <div class="num_example"> <h6 id="example_3">Example</h6> <p>The nature of the convergence condition in def. <a class="maruku-ref" href="#Convergence"></a> is well illuminated for instance by the <a class="existingWikiWord" href="/nlab/show/Serre+spectral+sequence">Serre</a>-<a class="existingWikiWord" href="/nlab/show/Atiyah-Hirzebruch+spectral+sequence">Atiyah-Hirzebruch spectral sequence</a>: for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">E^\bullet</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/generalized+cohomology+theory">generalized cohomology theory</a> and <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> a finite <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a>, then it converges to the <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>-cohomology of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, filtered by <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>-cohomology relative to the <a class="existingWikiWord" href="/nlab/show/simplicial+skeleton">skeleta</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>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mi>im</mi><mo stretchy="false">(</mo><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>X</mi> <mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>→</mo><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F^p E^{\bullet}(X) im( E^\bullet(X,X_{p-1}) \to E^\bullet(X))</annotation></semantics></math>. Moreover, the second page is the <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with coefficients in the <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>-ground ring, like so:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mi>E</mi> <mi>q</mi></msup><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⇒</mo><msup><mi>E</mi> <mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H^p(X,E^q(\ast)) \Rightarrow E^{p+q}(X) \,. </annotation></semantics></math></div> <p>Here the elements on the left in bidegree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p,q)</annotation></semantics></math> are manifestly given by cocycles that trivialize on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p-1)</annotation></semantics></math>-skeleton (being <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-cocycles), hence it is natural that these contribute to the filtering stage <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>E</mi> <mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mi>im</mi><mo stretchy="false">(</mo><msup><mi>E</mi> <mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>X</mi> <mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>→</mo><msup><mi>E</mi> <mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F^p E^{p+q}(X) = im(E^{p+q}(X,X_{p-1}) \to E^{p+q}(X))</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>In applications one is interested in computing the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">H_n</annotation></semantics></math> and uses spectral sequences converging to this as tools for approximating <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">H_n</annotation></semantics></math> in terms of the given filtration.</p> <p>Therefore usually spectral sequences are required to converge in each degree, or even that for each pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p,q)</annotation></semantics></math> there exists an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>r</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">r_0</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> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">r\geq r_0</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>d</mi> <mi>r</mi> <mrow><mi>p</mi><mo>−</mo><mi>r</mi><mo>,</mo><mi>q</mi><mo>+</mo><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msubsup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">d_r^{p-r,q+r-1} = 0</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>E</mi> <mi>r</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E^r)</annotation></semantics></math> collapses at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math>, then it converges to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">H_\bullet</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">H_n</annotation></semantics></math> being the unique entry <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>E</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow> <mi>r</mi></msubsup></mrow><annotation encoding="application/x-tex">E_{p,q}^r</annotation></semantics></math> on the non-vanishing row/column with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>+</mo><mi>q</mi><mo>=</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">p+q = n</annotation></semantics></math>.</p> </div> <h3 id="Boundedness">Boundedness</h3> <div class="num_defn" id="BoundedSpectralSequence"> <h6 id="definition_5">Definition</h6> <p>A spectral 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> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow> <mi>r</mi></msubsup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{E^r_{p,q}\}</annotation></semantics></math> is called a <strong>bounded spectral sequence</strong> if for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>,</mo><mi>r</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">n,r \in \mathbb{Z}</annotation></semantics></math> the number of non-vanishing terms of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>E</mi> <mrow><mi>k</mi><mo>,</mo><mi>n</mi><mo>−</mo><mi>k</mi></mrow> <mi>r</mi></msubsup></mrow><annotation encoding="application/x-tex">E^r_{k,n-k}</annotation></semantics></math> is finite.</p> </div> <div class="num_defn" id="QuadrantSpectralSequence"> <h6 id="example_4">Example</h6> <p>A <a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msubsup><mi>E</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow> <mi>r</mi></msubsup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{E^r_{p,q}\}</annotation></semantics></math> is called</p> <ul> <li> <p>a <strong>first quadrant spectral sequence</strong> if all terms except possibly for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>,</mo><mi>q</mi><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">p,q \geq 0</annotation></semantics></math> vanish;</p> </li> <li> <p>a <strong>third quadrant spectral sequence</strong> if all terms except possibly for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>,</mo><mi>q</mi><mo>≤</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">p,q \leq 0</annotation></semantics></math> vanish.</p> </li> </ul> <p>Such spectral sequences are bounded, def. <a class="maruku-ref" href="#BoundedSpectralSequence"></a>.</p> </div> <div class="num_prop" id="BoundedSpectralSequenceHasLimitTerm"> <h6 id="proposition">Proposition</h6> <p>A bounded spectral sequence, def. <a class="maruku-ref" href="#BoundedSpectralSequence"></a>, has a limit term, def. <a class="maruku-ref" href="#LimitTerm"></a>.</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>First notice that if a spectral sequence has at most <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> non-vanishing terms of total degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> on page <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math>, then all the following pages have at most at these positions non-vanishing terms, too, since these are the homologies of the previous terms.</p> <p>Therefore for a bounded spectral sequence for each <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> there is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">L(n) \in \mathbb{Z}</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>E</mi> <mrow><mi>p</mi><mo>,</mo><mi>n</mi><mo>−</mo><mi>p</mi></mrow> <mi>r</mi></msubsup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">E^r_{p,n-p} = 0</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>≤</mo><mi>L</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p \leq L(n)</annotation></semantics></math> and all <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>. Similarly there is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">T(n) \in \mathbb{Z}</annotation></semantics></math> such <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>E</mi> <mrow><mi>n</mi><mo>−</mo><mi>q</mi><mo>,</mo><mi>q</mi></mrow> <mi>r</mi></msubsup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">E^r_{n-q,q} = 0</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>≤</mo><mi>T</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q \leq T(n)</annotation></semantics></math> and all <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>We claim then that the limit term of the bounded spectral sequence is in position <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p,q)</annotation></semantics></math> given by the value <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>E</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow> <mi>r</mi></msubsup></mrow><annotation encoding="application/x-tex">E^r_{p,q}</annotation></semantics></math> for</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>></mo><mi>max</mi><mo stretchy="false">(</mo><mi>p</mi><mo>−</mo><mi>L</mi><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>,</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo>−</mo><mi>T</mi><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> r \gt max( p-L(p+q-1), q + 1 - T(p+q+1) ) \,. </annotation></semantics></math></div> <p>This is because for such <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> we have</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>E</mi> <mrow><mi>p</mi><mo>−</mo><mi>r</mi><mo>,</mo><mi>q</mi><mo>+</mo><mi>r</mi><mo>−</mo><mn>1</mn></mrow> <mi>r</mi></msubsup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">E^r_{p-r, q+r-1} = 0</annotation></semantics></math> because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>−</mo><mi>r</mi><mo><</mo><mi>L</mi><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p-r \lt L(p+q-1)</annotation></semantics></math>, and hence the <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ker</mi><mo stretchy="false">(</mo><msubsup><mo>∂</mo> <mrow><mi>p</mi><mo>−</mo><mi>r</mi><mo>,</mo><mi>q</mi><mo>+</mo><mi>r</mi><mo>−</mo><mn>1</mn></mrow> <mi>r</mi></msubsup><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">ker(\partial^r_{p-r,q+r-1}) = 0</annotation></semantics></math> vanishes;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>E</mi> <mrow><mi>p</mi><mo>+</mo><mi>r</mi><mo>,</mo><mi>q</mi><mo>−</mo><mi>r</mi><mo>+</mo><mn>1</mn></mrow> <mi>r</mi></msubsup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">E^r_{p+r, q-r+1} = 0</annotation></semantics></math> because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>−</mo><mi>r</mi><mo>+</mo><mn>1</mn><mo><</mo><mi>T</mi><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q-r + 1 \lt T(p+q+1)</annotation></semantics></math>, and hence the <a class="existingWikiWord" href="/nlab/show/image">image</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>im</mi><mo stretchy="false">(</mo><msubsup><mo>∂</mo> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow> <mi>r</mi></msubsup><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">im(\partial^r_{p,q}) = 0</annotation></semantics></math> vanishes.</p> </li> </ol> <p>Therefore</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> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow> <mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msubsup></mtd> <mtd><mo>=</mo><mi>ker</mi><mo stretchy="false">(</mo><msubsup><mo>∂</mo> <mrow><mi>p</mi><mo>−</mo><mi>r</mi><mo>,</mo><mi>q</mi><mo>+</mo><mi>r</mi><mo>−</mo><mn>1</mn></mrow> <mi>r</mi></msubsup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>im</mi><mo stretchy="false">(</mo><msubsup><mo>∂</mo> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow> <mi>r</mi></msubsup><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msubsup><mi>E</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow> <mi>r</mi></msubsup><mo stretchy="false">/</mo><mn>0</mn></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msubsup><mi>E</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow> <mi>r</mi></msubsup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} E^{r+1}_{p,q} &= ker(\partial^r_{p-r,q+r-1})/im(\partial^r_{p,q}) \\ & \simeq E^r_{p,q}/0 \\ & \simeq E^r_{p,q} \end{aligned} \,. </annotation></semantics></math></div></div> <h3 id="ExtensionProblem">Extension problem</h3> <p>Given a spectral sequence, then even if it converges strongly (def. <a class="maruku-ref" href="#Convergence"></a>), computing its infinity-page still just gives the <a class="existingWikiWord" href="/nlab/show/associated+graded">associated graded</a> of the <a class="existingWikiWord" href="/nlab/show/filtered+object">filtered object</a> that it converges to, not the filtered object itself. The latter is in each filter stage an <a class="existingWikiWord" href="/nlab/show/extension">extension</a> of the previous stage by the corresponding stage of the infinity-page, but there are in general several possible extensions (the trivial extension or some twisted extensions). The problem of determining these extensions and hence the problem of actually determining the filtered object from a spectral sequence converging to it is often referred to as the <em>extension problem</em>.</p> <p>More in detail, consider, for definiteness, a cohomology spectral sequence (def. <a class="maruku-ref" href="#CohomologySpectralSequence"></a>) converging (def. <a class="maruku-ref" href="#Convergence"></a>) to some <a class="existingWikiWord" href="/nlab/show/filtered+object">filtered</a> <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>H</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">F^\bullet H^\bullet</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msup><mspace width="thickmathspace"></mspace><mo>⇒</mo><mspace width="thickmathspace"></mspace><msup><mi>H</mi> <mo>•</mo></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> E^{p,q} \;\Rightarrow\; H^\bullet \,. </annotation></semantics></math></div> <p>Then by definition of convergence there are isomorphisms</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><mo>•</mo></mrow></msubsup><mo>≃</mo><msup><mi>F</mi> <mi>p</mi></msup><msup><mi>H</mi> <mrow><mi>p</mi><mo>+</mo><mo>•</mo></mrow></msup><mo stretchy="false">/</mo><msup><mi>F</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><msup><mi>H</mi> <mrow><mi>p</mi><mo>+</mo><mo>•</mo></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> E_\infty^{p,\bullet} \simeq F^p H^{p + \bullet} / F^{p+1} H^{p + \bullet} \,. </annotation></semantics></math></div> <p>Equivalently this means that there are <a class="existingWikiWord" href="/nlab/show/short+exact+sequences">short exact sequences</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><msup><mi>F</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><msup><mi>H</mi> <mrow><mi>p</mi><mo>+</mo><mo>•</mo></mrow></msup><mo>↪</mo><msup><mi>F</mi> <mi>p</mi></msup><msup><mi>H</mi> <mrow><mi>p</mi><mo>+</mo><mo>•</mo></mrow></msup><mo>⟶</mo><msubsup><mi>E</mi> <mn>∞</mn> <mrow><mi>p</mi><mo>,</mo><mo>•</mo></mrow></msubsup><mo>→</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> 0 \to F^{p+1}H^{p +\bullet} \hookrightarrow F^p H^{p +\bullet} \longrightarrow E_\infty^{p,\bullet} \to 0 \,. </annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>. The extension problem then is to inductively deduce <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>H</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">F^p H^\bullet</annotation></semantics></math> from knowledge of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>F</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><msup><mi>H</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">F^{p+1}H^\bullet</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>E</mi> <mn>∞</mn> <mrow><mi>p</mi><mo>,</mo><mo>•</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">E_\infty^{p,\bullet}</annotation></semantics></math>.</p> <p>In good cases these short exact sequences happen to be <a class="existingWikiWord" href="/nlab/show/split+exact+sequences">split exact sequences</a>, which means that the extension problem is solved by the <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a></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>H</mi> <mrow><mi>p</mi><mo>+</mo><mo>•</mo></mrow></msup><mo>≃</mo><msup><mi>F</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><msup><mi>H</mi> <mrow><mi>p</mi><mo>+</mo><mo>•</mo></mrow></msup><mo>⊕</mo><msubsup><mi>E</mi> <mn>∞</mn> <mrow><mi>p</mi><mo>,</mo><mo>•</mo></mrow></msubsup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> F^p H^{p+\bullet} \simeq F^{p+1} H^{p+\bullet} \oplus E_\infty^{p,\bullet} \,. </annotation></semantics></math></div> <p>But in general this need not be the case.</p> <p>One sufficient condition that these exact sequences split is that they consist of homomorphisms 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>-<a class="existingWikiWord" href="/nlab/show/modules">modules</a>, for some <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>, and that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>E</mi> <mn>∞</mn> <mrow><mi>p</mi><mo>,</mo><mo>•</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">E_\infty^{p,\bullet}</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/projective+modules">projective modules</a> (for instance <a class="existingWikiWord" href="/nlab/show/free+modules">free modules</a>) over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>. Because then the <a class="existingWikiWord" href="/nlab/show/Ext">Ext</a>-group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ext</mi> <mi>R</mi> <mn>1</mn></msubsup><mo stretchy="false">(</mo><msubsup><mi>E</mi> <mn>∞</mn> <mrow><mi>p</mi><mo>,</mo><mo>•</mo></mrow></msubsup><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ext^1_R(E_\infty^{p,\bullet},-)</annotation></semantics></math> vanishes, and hence all extensions are trivial, hence split.</p> <p>So for instance for every spectral sequence in <a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a> the extension problem is trivial (since every vector space is a free module).</p> <h2 id="Examples">Examples</h2> <p>The basic class of examples are</p> <ul> <li><a href="#SpectralSequenceOfFilteredComplex">Spectral sequences of filtered complexes</a></li> </ul> <p>which compute the cohomology of a <a class="existingWikiWord" href="/nlab/show/filtered+object">filtered complex</a> from the cohomologies of its <a class="existingWikiWord" href="/nlab/show/associated+graded+objects">associated graded objects</a>.</p> <p>From this one obtains as a special case the class of</p> <ul> <li><a href="#SpectralSequenceOfADoubleComplex">Spectral sequences of double complexes</a></li> </ul> <p>which compute the cohomology of the <a class="existingWikiWord" href="/nlab/show/total+complex">total complex</a> of a <a class="existingWikiWord" href="/nlab/show/double+complex">double complex</a> using the two canonical filtrations of this by row- and by column-degree.</p> <p>From this in turn one obtains as a special case the class of</p> <ul> <li><a href="#GrothendieckSpectralSequence">Grothendieck spectral sequences</a></li> </ul> <p>which compute the <a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>G</mi><mo>∘</mo><mi>F</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}^\bullet(G \circ F (-))</annotation></semantics></math> of the composite of two functors from the spectral sequence of the double complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mo>•</mo></msup><mi>G</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}^\bullet (F (\mathbb{R}^\bullet G (-)))</annotation></semantics></math>.</p> <p>Many special cases of this for various choices of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> and <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> go by special names, this we tabulate at</p> <ul> <li><em><a href="#ListOfExamples">List of examples</a></em>.</li> </ul> <h3 id="SpectralSequenceOfFilteredComplex">Spectral sequence of a filtered complex</h3> <p>The fundamental example of a spectral sequence, from which essentially all the other examples arise as special cases, is the <em><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+filtered+complex">spectral sequence of a filtered complex</a></em>. (See there for details). Or more generally in <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a>: the <a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+filtered+stable+homotopy+type">spectral sequence of a filtered stable homotopy type</a>.</p> <p>If a <a class="existingWikiWord" href="/nlab/show/cochain+complex">cochain complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">C^\bullet</annotation></semantics></math> is equipped with a <a class="existingWikiWord" href="/nlab/show/filtered+object">filtration</a> <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>, there is an induced filtration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>F</mi> <mo>•</mo></msup><mi>H</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F^\bullet H(C)</annotation></semantics></math> of its <a class="existingWikiWord" href="/nlab/show/cohomology+groups">cohomology groups</a>, according to which levels of the filtration contain representatives for the various cohomology classes.</p> <p>A filtration <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> also gives rise to an <a class="existingWikiWord" href="/nlab/show/associated+graded+object">associated graded object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Gr</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Gr(F)</annotation></semantics></math>, whose grades are the successive level inclusion <a class="existingWikiWord" href="/nlab/show/cokernels">cokernels</a>. Generically, the operations of grading and cohomology do not commute:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Gr</mi><mo stretchy="false">(</mo><msup><mi>F</mi> <mo>•</mo></msup><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≠</mo><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>Gr</mi><mo stretchy="false">(</mo><msup><mi>F</mi> <mo>•</mo></msup><mo stretchy="false">)</mo><mi>C</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Gr(F^\bullet H^\bullet(C)) \neq H^\bullet (Gr(F^\bullet) C) \,. </annotation></semantics></math></div> <p>But the <a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+filtered+complex">spectral sequence associated to a filtered complex</a> <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>, passes through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>Gr</mi><mo stretchy="false">(</mo><msup><mi>F</mi> <mo>⋆</mo></msup><mo stretchy="false">)</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^\bullet (Gr(F^\star) C)</annotation></semantics></math> in the page <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">E_{(1)}</annotation></semantics></math> and in good cases converges to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Gr</mi><mo stretchy="false">(</mo><msup><mi>F</mi> <mo>*</mo></msup><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Gr(F^* H^\bullet(C))</annotation></semantics></math>.</p> <h3 id="SpectralSequenceOfADoubleComplex">Spectral sequence of a double complex</h3> <p>The <a class="existingWikiWord" href="/nlab/show/total+complex">total complex</a> of a <a class="existingWikiWord" href="/nlab/show/double+complex">double complex</a> is naturally filtered in two ways: by columns and by rows. By the above <a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+filtered+complex">spectral sequence of a filtered complex</a> this gives two different spectral sequences associated computing the cohomology of a double complex from the cohomologies of its rows and columns. Many other classes of spectral sequences are special cases of this cases, notably the <a class="existingWikiWord" href="/nlab/show/Grothendieck+spectral+sequence">Grothendieck spectral sequence</a> and <em>its</em> special cases.</p> <p>This is discussed at <em><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+double+complex">spectral sequence of a double complex</a></em>.</p> <h3 id="HyperDerivedFunctors">Spectral sequences for hyper-derived functors</h3> <p>From the <a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+double+complex">spectral sequence of a double complex</a> one obtains as a special case a spectral sequence that computes <a class="existingWikiWord" href="/nlab/show/hyper-derived+functors">hyper-derived functors</a>.</p> <p>(…)</p> <h3 id="GrothendieckSpectralSequence">Grothendieck spectral sequence</h3> <p>The <em><a class="existingWikiWord" href="/nlab/show/Grothendieck+spectral+sequence">Grothendieck spectral sequence</a></em> computes the composite of two <a class="existingWikiWord" href="/nlab/show/derived+functors">derived functors</a> from the two derived functors separately.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mover><mo>→</mo><mi>F</mi></mover><mi>ℬ</mi><mover><mo>→</mo><mi>G</mi></mover><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{A} \stackrel{F}{\to} \mathcal{B} \stackrel{G}{\to} \mathcal{C}</annotation></semantics></math> be two <a class="existingWikiWord" href="/nlab/show/left+exact+functors">left exact functors</a> between <a class="existingWikiWord" href="/nlab/show/abelian+categories">abelian categories</a>.</p> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>R</mi> <mi>p</mi></msup><mi>F</mi><mo>:</mo><mi>𝒟</mi><mo>→</mo><mi>Ab</mi></mrow><annotation encoding="application/x-tex">R^p F : \mathcal{D} \to Ab</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a> of the <a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> etc. .</p> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> sends <a class="existingWikiWord" href="/nlab/show/injective+objects">injective objects</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/acyclic+objects">acyclic objects</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℬ</mi></mrow><annotation encoding="application/x-tex">\mathcal{B}</annotation></semantics></math> then for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">A \in \mathcal{A}</annotation></semantics></math> there is a <a href="#FirstQuadrant">first quadrant</a> cohomology spectral sequence</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> <mi>r</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup><mo>:</mo><mo>=</mo><mo stretchy="false">(</mo><msup><mi>R</mi> <mi>p</mi></msup><mi>G</mi><mo>∘</mo><msup><mi>R</mi> <mi>q</mi></msup><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> E_r^{p,q} := (R^p G \circ R^q F)(A) </annotation></semantics></math></div> <p>that converges to the right <a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a> of the composite functor</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> <mi>r</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup><mo>⇒</mo><msup><mi>R</mi> <mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></msup><mo stretchy="false">(</mo><mi>G</mi><mo>∘</mo><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> E_r^{p,q} \Rightarrow R^{p+q} (G \circ F)(A). </annotation></semantics></math></div> <p>Moreover</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/edge+maps">edge maps</a> in this spectral sequence are the canonical morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>R</mi> <mi>p</mi></msup><mi>G</mi><mo stretchy="false">(</mo><mi>F</mi><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>R</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mi>G</mi><mo>∘</mo><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> R^p G (F A) \to R^p (G \circ F)(A) </annotation></semantics></math></div> <p>induced from applying <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> to an injective resolution <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mover><mi>A</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">A \to \hat A</annotation></semantics></math> and the morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>R</mi> <mi>q</mi></msup><mo stretchy="false">(</mo><mi>G</mi><mo>∘</mo><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>G</mi><mo stretchy="false">(</mo><msup><mi>R</mi> <mi>q</mi></msup><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> R^q (G \circ F)(A) \to G(R^q F (A)) \,. </annotation></semantics></math></div></li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/exact+sequence">exact sequence</a> of low degree terms is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mo stretchy="false">(</mo><msup><mi>R</mi> <mn>1</mn></msup><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><msup><mi>R</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>G</mi><mo>∘</mo><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>G</mi><mo stretchy="false">(</mo><msup><mi>R</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><msup><mi>R</mi> <mn>2</mn></msup><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><msup><mi>R</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>G</mi><mo>∘</mo><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> 0 \to (R^1 G)(F(A)) \to R^1(G \circ F)(A) \to G(R^1(F(A))) \to (R^2 G)(F(A)) \to R^2(G \circ F)(A) </annotation></semantics></math></div></li> </ol> </div> <p>This is called the <em><a class="existingWikiWord" href="/nlab/show/Grothendieck+spectral+sequence">Grothendieck spectral sequence</a></em>.</p> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>Since for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mover><mi>A</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">A \to \hat A</annotation></semantics></math> an injective <a class="existingWikiWord" href="/nlab/show/resolution">resolution</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> the complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mover><mi>A</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(\hat A)</annotation></semantics></math> is a chain complex not concentrated in a single degree, we have that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>R</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mi>G</mi><mo>∘</mo><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R^p (G \circ F)(A)</annotation></semantics></math> is equivalently the <a class="existingWikiWord" href="/nlab/show/hyper-derived+functor">hyper-derived functor</a> evaluation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}^p(G) (F(A))</annotation></semantics></math>.</p> <p>Therefore the second spectral sequence discussed at <a href="#HyperDerivedFunctors">hyper-derived functor spectral sequences</a> converges as</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>R</mi> <mi>p</mi></msup><mi>G</mi><mo stretchy="false">)</mo><msup><mi>H</mi> <mi>q</mi></msup><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mover><mi>A</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⇒</mo><msup><mi>R</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mi>G</mi><mo>∘</mo><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (R^p G)H^q(F(\hat A)) \Rightarrow R^p (G \circ F)(A) \,. </annotation></semantics></math></div> <p>Now since by construction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>q</mi></msup><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mover><mi>A</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msup><mi>R</mi> <mi>q</mi></msup><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^q(F(\hat A)) = R^q F(A)</annotation></semantics></math> this is a spectral sequence</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>R</mi> <mi>p</mi></msup><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>R</mi> <mi>q</mi></msup><mi>F</mi><mo stretchy="false">)</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⇒</mo><msup><mi>R</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mi>G</mi><mo>∘</mo><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (R^p G)(R^q F) A) \Rightarrow R^p (G \circ F)(A) \,. </annotation></semantics></math></div> <p>This is the Grothendieck spectral sequence.</p> </div> <h3 id="SpecialGrothendieckSpectralSequence">Special Grothendieck spectral sequences</h3> <ul> <li> <p><a href="#LeraySpectralSequence">Leray spectral sequence</a></p> </li> <li> <p><a href="#HochschildSerreSpectralSequence">Hochschild-Serre spectral sequence</a></p> </li> <li> <p><a href="#BaseChangeSpectralSequence">Base-change spectral sequence</a></p> </li> </ul> <h4 id="LeraySpectralSequence">Leray spectral sequence</h4> <p>The <em><a class="existingWikiWord" href="/nlab/show/Leray+spectral+sequence">Leray spectral sequence</a></em> is the special case of the <a class="existingWikiWord" href="/nlab/show/Grothendieck+spectral+sequence">Grothendieck spectral sequence</a> for the case where the two functors being composed are a <a class="existingWikiWord" href="/nlab/show/direct+image">push-forward</a> of <a class="existingWikiWord" href="/nlab/show/sheaves">sheaves</a> of <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a> along a <a class="existingWikiWord" href="/nlab/show/continuous+map">continuous map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math> followed by the push-forward <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">X \to *</annotation></semantics></math> to the point. This yields a spectral sequence that computes the <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> on <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 terms of the abelian sheaf cohomology on <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>.</p> <div class="num_theorem"> <h6 id="theorem_2">Theorem</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X, Y</annotation></semantics></math> be suitable <a class="existingWikiWord" href="/nlab/show/sites">sites</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math> be a morphism of sites. () Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>=</mo><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Ab</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{C} = Ch_\bullet(Sh(X,Ab))</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi><mo>=</mo><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>Ab</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{D} = Ch_\bullet(Sh(Y,Ab))</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/model+structure+on+chain+complexes">model categories of complexes of sheaves of abelian groups</a>. The <a class="existingWikiWord" href="/nlab/show/direct+image">direct image</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">f_*</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/global+section">global section</a> functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>Y</mi></msub></mrow><annotation encoding="application/x-tex">\Gamma_Y</annotation></semantics></math> compose to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\Gamma_X</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>X</mi></msub><mo>:</mo><mi>𝒞</mi><mover><mo>→</mo><mrow><msub><mi>f</mi> <mo>*</mo></msub></mrow></mover><mi>𝒟</mi><mover><mo>→</mo><mrow><msub><mi>Γ</mi> <mi>Y</mi></msub></mrow></mover><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>Ab</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Gamma_X : \mathcal{C} \stackrel{f_*}{\to} \mathcal{D} \stackrel{\Gamma_Y}{\to} Ch_\bullet(Ab) \,. </annotation></semantics></math></div> <p>Then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Ab</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \in Sh(X,Ab)</annotation></semantics></math> a sheaf of abelian groups on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> there is a cohomology spectral sequence</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> <mi>r</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup><mo>:</mo><mo>=</mo><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msup><mi>R</mi> <mi>q</mi></msup><msub><mi>f</mi> <mo>*</mo></msub><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> E_r^{p,q} := H^p(Y, R^q f_* A) </annotation></semantics></math></div> <p>that converges as</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> <mi>r</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup><mo>⇒</mo><msup><mi>H</mi> <mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> E_r^{p,q} \Rightarrow H^{p+q}(X, A) </annotation></semantics></math></div> <p>and hence computes the cohomology of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with coefficients in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> in terms of the cohomology of <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> with coefficients in the push-forward of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>.</p> </div> <h4 id="BaseChangeSpectralSequence">Base change spectral sequence for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Tor</mi></mrow><annotation encoding="application/x-tex">Tor</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ext</mi></mrow><annotation encoding="application/x-tex">Ext</annotation></semantics></math></h4> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/ring">ring</a> write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Mod">Mod</a> for its category of <a class="existingWikiWord" href="/nlab/show/modules">modules</a>. Given a <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of <a class="existingWikiWord" href="/nlab/show/ring">ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msub><mi>R</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>R</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">f : R_1 \to R_2</annotation></semantics></math> and an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>R</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">R_2</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/module">module</a> <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> there are composites of <a class="existingWikiWord" href="/nlab/show/base+change">base change</a> along <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> with the <a class="existingWikiWord" href="/nlab/show/hom-functor">hom-functor</a> and the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>R</mi> <mn>1</mn></msub><mi>Mod</mi><mover><mo>→</mo><mrow><msub><mo>⊗</mo> <mrow><msub><mi>R</mi> <mn>1</mn></msub></mrow></msub><msub><mi>R</mi> <mn>2</mn></msub></mrow></mover><msub><mi>R</mi> <mn>2</mn></msub><mi>Mod</mi><mover><mo>→</mo><mrow><msub><mo>⊗</mo> <mrow><msub><mi>R</mi> <mn>2</mn></msub></mrow></msub><mi>N</mi></mrow></mover><mi>Ab</mi></mrow><annotation encoding="application/x-tex"> R_1 Mod \stackrel{\otimes_{R_1} R_2}{\to} R_2 Mod \stackrel{\otimes_{R_2} N}{\to} Ab </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>R</mi> <mn>1</mn></msub><mi>Mod</mi><mover><mo>→</mo><mrow><msub><mi>Hom</mi> <mrow><msub><mi>R</mi> <mn>1</mn></msub><mi>Mod</mi></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><msub><mi>R</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mover><msub><mi>R</mi> <mn>2</mn></msub><mi>Mod</mi><mover><mo>→</mo><mrow><msub><mi>Hom</mi> <mrow><msub><mi>R</mi> <mn>2</mn></msub></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>N</mi><mo stretchy="false">)</mo></mrow></mover><mi>Ab</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> R_1 Mod \stackrel{Hom_{R_1 Mod}(-,R_2)}{\to} R_2 Mod \stackrel{Hom_{R_2}(-,N)}{\to} Ab \,. </annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/derived+functors">derived functors</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mrow><msub><mi>R</mi> <mn>2</mn></msub></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>N</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom_{R_2}(-,N)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>⊗</mo> <mrow><msub><mi>R</mi> <mn>2</mn></msub></mrow></msub><mi>N</mi></mrow><annotation encoding="application/x-tex">\otimes_{R_2} N</annotation></semantics></math> are the <a class="existingWikiWord" href="/nlab/show/Ext">Ext</a>- and the <a class="existingWikiWord" href="/nlab/show/Tor">Tor</a>-functors, respectively, so the <a class="existingWikiWord" href="/nlab/show/Grothendieck+spectral+sequence">Grothendieck spectral sequence</a> applied to these composites yields <a class="existingWikiWord" href="/nlab/show/base+change+spectral+sequence">base change spectral sequence</a> for these.</p> <h4 id="HochschildSerreSpectralSequence">Hochschild-Serre spectral sequence</h4> <ul> <li><a class="existingWikiWord" href="/nlab/show/Hochschild-Serre+spectral+sequence">Hochschild-Serre spectral sequence</a></li> </ul> <h3 id="exact_couples">Exact couples</h3> <p>The above examples are all built on the <a href="#SpectralSequenceOfFilteredComplex">spectral sequence of a filtered complex</a>. An alternatively universal construction builds spectral sequences from <em>exact couples</em>.</p> <p>An <strong><a class="existingWikiWord" href="/nlab/show/exact+couple">exact couple</a></strong> is an exact sequence of three arrows among two objects</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><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><mo>.</mo></mrow><annotation encoding="application/x-tex"> E \overset{j}{\to} D \overset{\varphi}{\to} D \overset{k}{\to} E \overset{j}{\to}. </annotation></semantics></math></div> <p>These creatures construct spectral sequences by a two-step process:</p> <ul> <li>first, the composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>≔</mo><mi>k</mi><mi>j</mi><mo lspace="verythinmathspace">:</mo><mi>E</mi><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex"> d \coloneqq k j \colon E\to E</annotation></semantics></math> is nilpotent, in that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>d</mi> <mn>2</mn></msup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">d^2=0</annotation></semantics></math></li> <li>second, the homology <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">E'</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E,d)</annotation></semantics></math> supports a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo>′</mo><mo>:</mo><mi>E</mi><mo>′</mo><mo>→</mo><mi>φ</mi><mi>D</mi></mrow><annotation encoding="application/x-tex">j':E'\to \varphi D</annotation></semantics></math>, and receives a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>′</mo><mo>:</mo><mi>φ</mi><mi>D</mi><mo>→</mo><mi>E</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">k':\varphi D\to E'</annotation></semantics></math>. Setting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>′</mo><mo>=</mo><mi>φ</mi><mi>D</mi></mrow><annotation encoding="application/x-tex">D'=\varphi D</annotation></semantics></math>, by general reasoning</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>′</mo><mover><mo>→</mo><mrow><mi>j</mi><mo>′</mo></mrow></mover><mi>D</mi><mo>′</mo><mover><mo>→</mo><mi>φ</mi></mover><mi>D</mi><mo>′</mo><mover><mo>→</mo><mrow><mi>k</mi><mo>′</mo></mrow></mover><mi>E</mi><mo>′</mo><mover><mo>→</mo><mrow><mi>j</mi><mo>′</mo></mrow></mover><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> E' \overset{j'}{\to} D' \overset{\varphi}{\to} D' \overset{k'}{\to} E' \overset{j'}{\to} \,. </annotation></semantics></math></div> <p>is again an exact couple.</p> <p>The sequence of complexes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>E</mi><mo>′</mo><mo>,</mo><mi>d</mi><mo>′</mo><mo stretchy="false">)</mo><mo>,</mo><mi>…</mi></mrow><annotation encoding="application/x-tex">(E,d),(E',d'),\dots</annotation></semantics></math> is a spectral sequence, by construction. For more see at <em><a href="exact+couple#SpectralSequencesFromExactCouples">exact couple – Spectral sequences from exact couples</a></em></p> <p>Examples of exact couples can be constructed in a number of ways. Notably there are naturally <a href="exact+couple#ExactCoupleOfATowerOfFibrations">exact couples of towers of (co-)fibrations</a>. For instance <a class="existingWikiWord" href="/nlab/show/Adams+spectral+sequences">Adams spectral sequences</a> are usually produced this way (from towers which are <a class="existingWikiWord" href="/nlab/show/Adams+resolutions">Adams resolutions</a>). For a list of examples in this class see <a href="#SpectralSequencesInducedFromTowersOfSpectra">below</a>.</p> <p>Importantly, any short exact sequence involving two distinct chain complexes provides an exact couple among their total homology complexes, via the Mayer-Vietoris long exact sequence; in particular, applying this procedure to the relative homology of a filtered complex gives precisely the spectral sequence of the filtered complex described (???) somewhere else on this page. For another example, choosing a chain complex of flat modules <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>C</mi> <mover><mo>,</mo><mo>˙</mo></mover></msup><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C^\dot,d)</annotation></semantics></math>, tensoring with the short exact sequence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mi>p</mi><mi>ℤ</mi><mo>→</mo><mi>ℤ</mi><mo stretchy="false">/</mo><msup><mi>p</mi> <mn>2</mn></msup><mi>ℤ</mi><mo>→</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mi>p</mi><mi>ℤ</mi></mrow><annotation encoding="application/x-tex"> \mathbb{Z}/p\mathbb{Z} \to \mathbb{Z}/p^2\mathbb{Z} \to \mathbb{Z}/p\mathbb{Z} </annotation></semantics></math></div> <p>gives the exact couple</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">/</mo><msup><mi>p</mi> <mn>2</mn></msup><mi>ℤ</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mo stretchy="false">[</mo><mo>⋅</mo><mo stretchy="false">]</mo></mrow></mover><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mi>p</mi><mi>ℤ</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mi>β</mi></mover><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mi>p</mi><mi>ℤ</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mi>p</mi></mover><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">/</mo><msup><mi>p</mi> <mn>2</mn></msup><mi>ℤ</mi><mo stretchy="false">)</mo><mi>⋯</mi></mrow><annotation encoding="application/x-tex"> H^\bullet(d,\mathbb{Z}/p^2\mathbb{Z}) \overset{[\cdot]}{\to} H^\bullet(d,\mathbb{Z}/p\mathbb{Z}) \overset{\beta}{\to} H^\bullet(d,\mathbb{Z}/p\mathbb{Z}) \overset{p}{\to}H^\bullet(d,\mathbb{Z}/p^2\mathbb{Z})\cdots </annotation></semantics></math></div> <p>in which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math> is the <em>mod-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> Bockstein</em> homomorphism.</p> <p>The exact couple recipe for spectral sequences is notable in that it doesn’t mention any grading on the objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>,</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">D,E</annotation></semantics></math>; trivially, an exact couple can be specified by a short exact sequence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">coker</mo><mi>φ</mi><mo>→</mo><mi>E</mi><mo>→</mo><mi>ker</mi><mi>φ</mi></mrow><annotation encoding="application/x-tex">\coker \varphi\to E\to \ker\varphi</annotation></semantics></math>, although this obscures the focus usually given to <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>. In applications, a bi-grading is usually induced by the context, which also specifies bidegrees for the initial maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>φ</mi></mrow><annotation encoding="application/x-tex">j,k,\varphi</annotation></semantics></math>, leading to the conventions mentioned earlier.</p> <h3 id="ListOfExamples">List of examples</h3> <div> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/tower+diagram">tower diagram</a>/<a class="existingWikiWord" href="/nlab/show/filtered+object+in+an+%28infinity%2C1%29-category">filtering</a></th><th><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+filtered+stable+homotopy+type">spectral sequence of a filtered stable homotopy type</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/filtered+chain+complex">filtered chain complex</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+filtered+complex">spectral sequence of a filtered complex</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Postnikov+tower">Postnikov tower</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Atiyah-Hirzebruch+spectral+sequence">Atiyah-Hirzebruch spectral sequence</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/chromatic+tower">chromatic tower</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/chromatic+spectral+sequence">chromatic spectral sequence</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/simplicial+skeleton">skeleta</a> of <a class="existingWikiWord" href="/nlab/show/simplicial+object+in+an+%28%E2%88%9E%2C1%29-category">simplicial object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+simplicial+stable+homotopy+type">spectral sequence of a simplicial stable homotopy type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/simplicial+skeleton">skeleta</a> of <a class="existingWikiWord" href="/nlab/show/Sweedler+coring">Sweedler coring</a> of <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+algebra">E-∞ algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Adams+spectral+sequence">Adams spectral sequence</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/filtration+by+support">filtration by support</a></td><td style="text-align: left;">…</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/slice+filtration">slice filtration</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/slice+spectral+sequence">slice spectral sequence</a></td></tr> </tbody></table> </div> <p>The following list of examples orders the various classes of spectral sequences by special cases: items further to the right are special cases of items further to the left.</p> <ul> <li> <p><strong><em><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+filtered+complex">spectral sequence of a filtered complex</a></em></strong> (approximate <a class="existingWikiWord" href="/nlab/show/chain+homology">chain homology</a> by higher-order <a class="existingWikiWord" href="/nlab/show/relative+homology">relative homology</a> in the presence of a <a class="existingWikiWord" href="/nlab/show/filtered+object">filtering</a>)</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah-Hirzebruch+spectral+sequence">Atiyah-Hirzebruch spectral sequence</a> (compute <a class="existingWikiWord" href="/nlab/show/generalized+homology">generalized homology</a> of a <a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre fibration</a> in terms of that of the base with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in that of the <a class="existingWikiWord" href="/nlab/show/fibers">fibers</a>)</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Serre+spectral+sequence">Serre spectral sequence</a> (compute <a class="existingWikiWord" href="/nlab/show/ordinary+homology">ordinary homology</a> of a <a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre fibration</a> in terms of that of the base with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in that of the <a class="existingWikiWord" href="/nlab/show/fibers">fibers</a>)</li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eilenberg-Moore+spectral+sequence">Eilenberg-Moore spectral sequence</a> (compute homology of <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+products">homotopy fiber products</a>)</p> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+double+complex">spectral sequence of a double complex</a></strong> (compute homology of <a class="existingWikiWord" href="/nlab/show/total+complex">total complex</a> by filtering by row/column degree)</p> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/Grothendieck+spectral+sequence">Grothendieck spectral sequence</a></strong> (compute <a class="existingWikiWord" href="/nlab/show/composition">composition</a> of two <a class="existingWikiWord" href="/nlab/show/derived+functors+in+homological+algebra">derived functors</a>)</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/base+change+spectral+sequence">base change spectral sequence</a> (compute <a class="existingWikiWord" href="/nlab/show/base+change">base change</a>/<a class="existingWikiWord" href="/nlab/show/extension+of+scalars">extension of scalars</a> in two stages)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Leray+spectral+sequence">Leray spectral sequence</a> (compute <a class="existingWikiWord" href="/nlab/show/global+sections">global sections</a> in two stages)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild-Serre+spectral+sequence">Hochschild-Serre spectral sequence</a> (compute <a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a> in two stages)</p> </li> </ul> </li> </ul> </li> </ul> </li> </ul> <p>Here is a more random list (using material from <a href="#Wikipedia">Wikipedia</a>). Eventually to be merged with the above.</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Adams+spectral+sequence">Adams spectral sequence</a> in <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Adams-Novikov+spectral+sequence">Adams-Novikov spectral sequence</a>, converging to <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> of <a class="existingWikiWord" href="/nlab/show/connective+spectra">connective spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chromatic+spectral+sequence">chromatic spectral sequence</a> for calculating the initial terms of the <a class="existingWikiWord" href="/nlab/show/Adams-Novikov+spectral+sequence">Adams-Novikov spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah-Hirzebruch+spectral+sequence">Atiyah-Hirzebruch spectral sequence</a> of an <a class="existingWikiWord" href="/nlab/show/extraordinary+cohomology+theory">extraordinary cohomology theory</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/descent+spectral+sequence">descent spectral sequence</a></p> </li> <li> <p><span class="newWikiWord">Bar spectral sequence<a href="/nlab/new/Bar+spectral+sequence">?</a></span> for the <a class="existingWikiWord" href="/nlab/show/homology">homology</a> of the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> of a <a class="existingWikiWord" href="/nlab/show/group">group</a></p> </li> <li> <p><span class="newWikiWord">Barratt spectral sequence<a href="/nlab/new/Barratt+spectral+sequence">?</a></span> converging to the <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> of the initial space of a <a class="existingWikiWord" href="/nlab/show/cofibration">cofibration</a></p> </li> <li> <p><span class="newWikiWord">Bloch-Lichtenbaum spectral sequence<a href="/nlab/new/Bloch-Lichtenbaum+spectral+sequence">?</a></span> converging to the <a class="existingWikiWord" href="/nlab/show/algebraic+K-theory">algebraic K-theory</a> of a <a class="existingWikiWord" href="/nlab/show/field">field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bockstein+spectral+sequence">Bockstein spectral sequence</a> relating the <a class="existingWikiWord" href="/nlab/show/homology">homology</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>mod</mi><mi>p</mi></mrow><annotation encoding="application/x-tex">mod p</annotation></semantics></math> coefficients and the homology reduced <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>mod</mi><mi>p</mi></mrow><annotation encoding="application/x-tex">mod p</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bousfield-Kan+spectral+sequence">Bousfield-Kan spectral sequence</a> converging to the <a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a> of a <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> <li> <p><span class="newWikiWord">Cartan-Leray spectral sequence<a href="/nlab/new/Cartan-Leray+spectral+sequence">?</a></span> converging to the <a class="existingWikiWord" href="/nlab/show/homology">homology</a> of a <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a></p> </li> <li> <p><span class="newWikiWord">Čech-to-derived functor spectral sequence<a href="/nlab/new/%C4%8Cech-to-derived+functor+spectral+sequence">?</a></span> from <a class="existingWikiWord" href="/nlab/show/%C4%8Cech+cohomology">Čech cohomology</a> to <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> </li> <li> <p><span class="newWikiWord">change of rings spectral sequences<a href="/nlab/new/change+of+rings+spectral+sequences">?</a></span> for calculating <a class="existingWikiWord" href="/nlab/show/Tor">Tor</a> and <a class="existingWikiWord" href="/nlab/show/Ext">Ext</a> groups of modules</p> </li> <li> <p><span class="newWikiWord">Connes spectral sequences<a href="/nlab/new/Connes+spectral+sequences">?</a></span> converging to the <a class="existingWikiWord" href="/nlab/show/cyclic+homology">cyclic homology</a> of an algebra</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/EHP+spectral+sequence">EHP spectral sequence</a> converging to <a class="existingWikiWord" href="/nlab/show/stable+homotopy+groups+of+spheres">stable homotopy groups of spheres</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eilenberg-Moore+spectral+sequence">Eilenberg-Moore spectral sequence</a> for the <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a> of the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> of a <a class="existingWikiWord" href="/nlab/show/fibration">fibration</a></p> </li> <li> <p><span class="newWikiWord">Federer spectral sequence<a href="/nlab/new/Federer+spectral+sequence">?</a></span> converging to <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> of a function space</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fr%C3%B6licher+spectral+sequence">Frölicher spectral sequence</a> starting from the <a class="existingWikiWord" href="/nlab/show/Dolbeault+cohomology">Dolbeault cohomology</a> and converging to the algebraic <a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a> of a <a class="existingWikiWord" href="/nlab/show/variety">variety</a></p> </li> <li> <p><span class="newWikiWord">Green's spectral sequence<a href="/nlab/new/Green%27s+spectral+sequence">?</a></span> for <span class="newWikiWord">Koszul cohomology<a href="/nlab/new/Koszul+cohomology">?</a></span></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+spectral+sequence">Grothendieck spectral sequence</a> for composing <a class="existingWikiWord" href="/nlab/show/derived+functors">derived functors</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge-de+Rham+spectral+sequence">Hodge-de Rham spectral sequence</a> converging to the algebraic <a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a> of a <a class="existingWikiWord" href="/nlab/show/variety">variety</a></p> </li> <li> <p><span class="newWikiWord">Hurewicz spectral sequence<a href="/nlab/new/Hurewicz+spectral+sequence">?</a></span> for calculating the <a class="existingWikiWord" href="/nlab/show/homology">homology</a> of a space from its <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a></p> </li> <li> <p><span class="newWikiWord">hyperhomology spectral sequence<a href="/nlab/new/hyperhomology+spectral+sequence">?</a></span> for calculating <span class="newWikiWord">hyperhomology<a href="/nlab/new/hyperhomology">?</a></span></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+spectral+sequence">Künneth spectral sequence</a> for calculating the homology of a <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> of <a class="existingWikiWord" href="/nlab/show/differential+graded+algebras">differential graded algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Leray+spectral+sequence">Leray spectral sequence</a> converging to <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Leray-Serre+spectral+sequence">Leray-Serre spectral sequence</a> of a <a class="existingWikiWord" href="/nlab/show/fibration">fibration</a></p> </li> <li> <p><span class="newWikiWord">Lyndon-Hochschild?Serre spectral sequence<a href="/nlab/new/Lyndon-Hochschild%3FSerre+spectral+sequence">?</a></span> in <a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/May+spectral+sequence">May spectral sequence</a> for calculating the <a class="existingWikiWord" href="/nlab/show/Tor">Tor</a> or <a class="existingWikiWord" href="/nlab/show/Ext">Ext</a> groups of an algebra</p> </li> <li> <p><span class="newWikiWord">Miller spectral sequence<a href="/nlab/new/Miller+spectral+sequence">?</a></span> converging to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>mod</mi><mi>p</mi></mrow><annotation encoding="application/x-tex">mod p</annotation></semantics></math> <span class="newWikiWord">stable homology<a href="/nlab/new/stable+homology">?</a></span> of a space</p> </li> <li> <p><span class="newWikiWord">Quillen spectral sequence<a href="/nlab/new/Quillen+spectral+sequence">?</a></span> for calculating the homotopy of a <a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+an+exact+couple">spectral sequence of an exact couple</a></p> </li> <li> <p><span class="newWikiWord">van Est spectral sequence<a href="/nlab/new/van+Est+spectral+sequence">?</a></span> converging to relative <a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a></p> </li> <li> <p><span class="newWikiWord">van Kampen spectral sequence<a href="/nlab/new/van+Kampen+spectral+sequence">?</a></span> for calculating the homotopy of a wedge of spaces.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/slice+spectral+sequence">slice spectral sequence</a></p> </li> </ul> <h2 id="properties">Properties</h2> <h3 id="basic_lemmas">Basic lemmas</h3> <div class="num_lemma"> <h6 id="lemma">Lemma</h6> <p><strong>(mapping lemma)</strong></p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mo stretchy="false">(</mo><msubsup><mi>E</mi> <mi>r</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup><mo>→</mo><mo stretchy="false">(</mo><msubsup><mi>F</mi> <mi>r</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f : (E_r^{p,q} \to (F_r^{p,q}))</annotation></semantics></math> is a morphism of spectral sequences such that for some <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> we have that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>r</mi></msub><mo>:</mo><msubsup><mi>E</mi> <mi>r</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup><msubsup><mo lspace="0em" rspace="thinmathspace">toF</mo> <mi>r</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">f_r : E_r^{p,q} \toF_r^{p,q}</annotation></semantics></math> is an isomorphism, then also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>s</mi></msub></mrow><annotation encoding="application/x-tex">f_s</annotation></semantics></math> is an isomorphism for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>≥</mo><mi>r</mi></mrow><annotation encoding="application/x-tex">s \geq r</annotation></semantics></math>.</p> </div> <div class="num_lemma"> <h6 id="lemma_2">Lemma</h6> <p><strong>(classical convergence theorem)</strong></p> <p>(…)</p> </div> <p>This is recalled in (<a href="#Weibel">Weibel, theorem 5.51</a>).</p> <h3 id="FirstQuadrant">First quadrant spectral sequence</h3> <div class="num_defn"> <h6 id="definition_6">Definition</h6> <p>A <strong>first quadrant spectral sequence</strong> is one for wich all pages are concentrated in the first quadrant of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p,q)</annotation></semantics></math>-plane, in that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>p</mi><mo><</mo><mn>0</mn><mo stretchy="false">)</mo><mi>or</mi><mo stretchy="false">(</mo><mi>q</mi><mo><</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⇒</mo><msubsup><mi>E</mi> <mi>r</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> ((p \lt 0) or (q \lt 0)) \;\; \Rightarrow E_r^{p,q} = 0 \,. </annotation></semantics></math></div></div> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>If the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math>th page is concentrated in the first quadrant, then so the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>r</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mi>st</mi></mrow><annotation encoding="application/x-tex">(r+1)st</annotation></semantics></math> page. So if the first one is, then all are.</p> </div> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>Every first quadrant spectral sequence converges at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p,q)</annotation></semantics></math> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>></mo><mi>max</mi><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">r \gt max(p,q+1)</annotation></semantics></math> on</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> <mrow><mi>max</mi><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mn>1</mn></mrow> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup><mo>=</mo><msubsup><mi>E</mi> <mn>∞</mn> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> E_{max(p,q+1)+1}^{p,q} = E_\infty^{p,q} \,. </annotation></semantics></math></div></div> <div class="num_prop"> <h6 id="proposition_4">Proposition</h6> <p>If a first quadrant spectral sequence converges</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> <mi>r</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup><mo>⇒</mo><msup><mi>H</mi> <mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></msup></mrow><annotation encoding="application/x-tex"> E_r^{p,q} \Rightarrow H^{p+q} </annotation></semantics></math></div> <p>then each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">H^n</annotation></semantics></math> has a filtration of length <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>=</mo><msup><mi>F</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><msup><mi>H</mi> <mi>n</mi></msup><mo>⊂</mo><msup><mi>F</mi> <mi>n</mi></msup><msup><mi>H</mi> <mi>n</mi></msup><mo>⊂</mo><mi>⋯</mi><mo>⊂</mo><msup><mi>F</mi> <mn>1</mn></msup><msup><mi>H</mi> <mi>n</mi></msup><mo>⊂</mo><msup><mi>F</mi> <mn>0</mn></msup><msup><mi>H</mi> <mi>n</mi></msup><mo>=</mo><msup><mi>H</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex"> 0 = F^{n+1}H^n \subset F^n H^n \subset \cdots \subset F^1 H^n \subset F^0 H^n = H^n </annotation></semantics></math></div> <p>and we have</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>F</mi> <mi>n</mi></msup><msup><mi>H</mi> <mi>n</mi></msup><mo>≃</mo><msubsup><mi>E</mi> <mn>∞</mn> <mrow><mi>n</mi><mo>,</mo><mn>0</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">F^n H^n \simeq E_\infty^{n,0}</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">/</mo><msup><mi>F</mi> <mn>1</mn></msup><msup><mi>H</mi> <mi>n</mi></msup><mo>≃</mo><msubsup><mi>E</mi> <mn>∞</mn> <mrow><mn>0</mn><mo>,</mo><mi>n</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">H^n/F^1 H^n \simeq E_\infty^{0,n}</annotation></semantics></math>.</p> </li> </ul> </div> <h3 id="PropertiesCupProductStructure">Cup product structure</h3> <p>Cohomological spectral sequences are compatible with <a class="existingWikiWord" href="/nlab/show/cup+product">cup product</a> structure on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">E_2</annotation></semantics></math>-page. (e.g. <a href="#Hutchings11">Hutchings 11, sections 5 and 6</a>)</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/edge+morphism">edge morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multiplicative+spectral+sequence">multiplicative spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequences+in+homotopy+type+theory">spectral sequences in homotopy type theory</a></p> </li> </ul> <h2 id="references">References</h2> <p>Spectral sequences were originally introduced in 1946 by <a class="existingWikiWord" href="/nlab/show/Jean+Leray">Jean Leray</a> in the paper</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jean+Leray">Jean Leray</a>, <em>Structure de l’anneau d’homologie d’une représentation</em>, Comptes rendus hebdomadaires des séances de l’Académie des Sciences 222 (1946) 1419–1422.</li> </ul> <h3 id="general_exposition">General exposition</h3> <ul> <li><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</li> </ul> <h3 id="abelianstable_theory">Abelian/stable theory</h3> <p>An elementary pedagogical introduction is in</p> <ul id="Chow"> <li><a class="existingWikiWord" href="/nlab/show/Timothy+Chow">Timothy Chow</a>, <em>You could have invented spectral sequences</em>, Notices of the AMS (2006) (<a href="http://www.ams.org/notices/200601/fea-chow.pdf">pdf</a>)</li> </ul> <p>Textbook accounts:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/John+McCleary">John McCleary</a>: <em>A User’s Guide to Spectral Sequences</em>, Cambridge University Press (2010) [<a href="https://doi.org/10.1017/CBO9780511626289">doi:10.1017/CBO9780511626289</a>]</p> </li> <li id="Weibel"> <p><a class="existingWikiWord" href="/nlab/show/Charles+Weibel">Charles Weibel</a>, chapter 5 of: <em>An introduction to homological algebra</em> Cambridge studies in advanced mathematics 38 (1994)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Raoul+Bott">Raoul Bott</a>, <a class="existingWikiWord" href="/nlab/show/Loring+Tu">Loring Tu</a>, section 14 of <em>Differential forms in algebraic topology</em>, Graduate Texts in Mathematics <strong>82</strong>, Springer 1982. xiv+331 pp.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hal+Schenck">Hal Schenck</a>, <em>Chapter 9: Cohomology and spectral sequences</em> (<a href="http://www.math.uiuc.edu/~schenck/tapp.pdf">pdf</a>) .</p> </li> <li id="Hatcher"> <p><a class="existingWikiWord" href="/nlab/show/Alan+Hatcher">Alan Hatcher</a>, <em>Spectral sequences in algebraic topology</em> (<a href="http://www.math.cornell.edu/~hatcher/SSAT/SSATpage.html">web</a>)</p> </li> <li id="Kochmann96"> <p><a class="existingWikiWord" href="/nlab/show/Stanley+Kochmann">Stanley Kochmann</a>, <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> <p><a class="existingWikiWord" href="/nlab/show/Dai+Tamaki">Dai Tamaki</a>, <a class="existingWikiWord" href="/nlab/show/Akira+Kono">Akira Kono</a>, Chapter 5 in: <em>Generalized Cohomology</em>, Translations of Mathematical Monographs, American Mathematical Society, 2006 (<a href="https://bookstore.ams.org/mmono-230">ISBN: 978-0-8218-3514-2</a>)</p> </li> </ul> <p>The general discussion in the context of <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a> theory (the <a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+filtered+stable+homotopy+type">spectral sequence of a filtered stable homotopy type</a>) is in section 1.2.2 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/Higher+Algebra">Higher Algebra</a></em></li> </ul> <p>A review Master thesis is</p> <ul> <li>Jennifer Orlich, <em>Spectral sequences and an application</em> (<a href="http://www.math.osu.edu/~flicker.1/orlich.pdf">pdf</a>)</li> </ul> <p>Reviews of and lecture notes on standard definitions and facts about spectral sequences include</p> <ul> <li> <p>Matthew Greenberg, <em>Spectral sequences</em> (<a href="http://www.math.mcgill.ca/goren/SeminarOnCohomology/specseq.pdf">pdf</a>)</p> </li> <li id="Hutchings11"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Hutchings">Michael Hutchings</a>, <em>Introduction to spectral sequences</em> (<a href="http://math.berkeley.edu/~hutching/teach/215b-2011/ss.pdf">pdf</a>)</p> </li> <li> <p>Daniel Murfet, <em>Spectral sequences</em> (<a href="http://therisingsea.org/notes/SpectralSequences.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Neil+Strickland">Neil Strickland</a>, <em>Spectral sequences</em> (<a href="http://neil-strickland.staff.shef.ac.uk/courses/bestiary/ss.pdf">pdf</a>)</p> </li> <li> <p>Ravi Vakil, <em>Spectral Sequences: Friend or Foe?</em> (<a href="http://math.stanford.edu/~vakil/0708-216/216ss.pdf">pdf</a>)</p> </li> <li> <p>Brandon Williams, <em>Spectral sequences</em> (<a href="http://www.math.sunysb.edu/~mbw/notes/orals/Spectral%20Sequences.pdf">pdf</a>)</p> </li> <li id="Boardman99"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Boardman">Michael Boardman</a>, <em>Conditionally convergent spectral sequences</em>, 1999 (<a href="http://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/boardman-conditionally-1999.pdf">pdf</a>)</p> </li> </ul> <p>See also</p> <ul id="Wikipedia"> <li>Wikipedia, <em><a href="http://en.wikipedia.org/wiki/Spectral_sequence">Spectral sequence</a></em></li> </ul> <ul> <li> <p>A. Romero, J. Rubio, F. Sergeraert, <em>Computing spectral sequences</em> (<a href="http://www-fourier.ujf-grenoble.fr/~sergerar/Papers/Ana-JSC.pdf">pdf</a>)</p> </li> <li> <p>Eric Peterson, <em><a href="http://ext-chart.org">Ext chart</a></em> software for computing spectral sequences</p> </li> </ul> <h3 id="ReferencesNonabelian">Nonabelian / unstable theory</h3> <p>Homotopy spectral sequences in model categories are discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Aldridge+Bousfield">Aldridge Bousfield</a>, <em>Cosimplicial resolutions and homotopy spectral sequences in model categories</em> (<a href="http://arxiv.org/abs/math/0312531">arXiv:math/0312531</a>).</li> </ul> <p>Spectral sequences in general categories with <a class="existingWikiWord" href="/nlab/show/zero+morphisms">zero morphisms</a> are discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Marco+Grandis">Marco Grandis</a>, <em>Homotopy spectral sequences</em> (<a href="http://arxiv.org/abs/1007.0632">arXiv:1007.0632</a>)</li> </ul> <p>Discussion from a perspective of <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> (see also <em><a class="existingWikiWord" href="/nlab/show/spectral+sequences+in+homotopy+type+theory">spectral sequences in homotopy type theory</a></em>):</p> <ul> <li id="Shulman13"> <p><a class="existingWikiWord" href="/nlab/show/Mike+Shulman">Mike Shulman</a>, <em><a class="existingWikiWord" href="/homotopytypetheory/show/spectral+sequences">Spectral sequences</a></em> 2013 (<a href="https://golem.ph.utexas.edu/category/2013/08/what_is_a_spectral_sequence.html">part I</a>, <a href="http://homotopytypetheory.org/2013/08/08/spectral-sequences/">part II</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Floris+van+Doorn">Floris van Doorn</a>, §5 in: <em>On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory</em> (2018) [<a href="https://arxiv.org/abs/1808.10690">arXiv:1808.10690</a>]</p> </li> </ul> <p>and implementation in <a class="existingWikiWord" href="/nlab/show/Lean">Lean</a>-<a class="existingWikiWord" href="/nlab/show/HoTT">HoTT</a> is in</p> <ul> <li id="vanDoornSpectral"><a class="existingWikiWord" href="/nlab/show/Floris+van+Doorn">Floris van Doorn</a>, <a class="existingWikiWord" href="/nlab/show/Egbert+Rijke">Egbert Rijke</a>, <a class="existingWikiWord" href="/nlab/show/Ulrik+Buchholtz">Ulrik Buchholtz</a>, <a class="existingWikiWord" href="/nlab/show/Favonia">Favonia</a>, <a class="existingWikiWord" href="/nlab/show/Steve+Awodey">Steve Awodey</a>, <a class="existingWikiWord" href="/nlab/show/Jeremy+Avigad">Jeremy Avigad</a>, <a class="existingWikiWord" href="/nlab/show/Mike+Shulman">Mike Shulman</a>, <a class="existingWikiWord" href="/nlab/show/Jonas+Frey">Jonas Frey</a>, <em>Spectral</em> [<a href="https://github.com/cmu-phil/Spectral">github.com/cmu-phil/Spectral</a>]</li> </ul> <h3 id="history">History</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/John+McCleary">John McCleary</a>, <em>A history of spectral sequences: Origins to 1953</em>, in <em>History of Topology</em>, edited by Ioan M. James, North Holland (1999) 631–663</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 21, 2024 at 07:43:56. See the <a href="/nlab/history/spectral+sequence" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/spectral+sequence" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/2161/#Item_34">Discuss</a><span class="backintime"><a href="/nlab/revision/spectral+sequence/76" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/spectral+sequence" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/spectral+sequence" accesskey="S" class="navlink" id="history" rel="nofollow">History (76 revisions)</a> <a href="/nlab/show/spectral+sequence/cite" style="color: black">Cite</a> <a href="/nlab/print/spectral+sequence" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/spectral+sequence" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>