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> Grothendieck 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> Grothendieck 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/4450/#Item_1" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="homological_algebra">Homological algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a></strong></p> <p>(also <a class="existingWikiWord" href="/nlab/show/nonabelian+homological+algebra">nonabelian homological algebra</a>)</p> <p><em><a class="existingWikiWord" href="/schreiber/show/Introduction+to+Homological+Algebra">Introduction</a></em></p> <p><strong>Context</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+and+abelian+categories">additive and abelian categories</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ab-enriched+category">Ab-enriched category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-additive+category">pre-additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-abelian+category">pre-abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+category">Grothendieck category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaves">abelian sheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semi-abelian+category">semi-abelian category</a></p> </li> </ul> <p><strong>Basic definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/kernel">kernel</a>, <a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex">complex</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential">differential</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homology">homology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+map">chain map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+homotopy">chain homotopy</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+homology+and+cohomology">chain homology and cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+resolution">homological resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+on+a+simplicial+set">simplicial homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+homology">generalized homology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exact+sequence">exact sequence</a>,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a>, <a class="existingWikiWord" href="/nlab/show/long+exact+sequence">long exact sequence</a>, <a class="existingWikiWord" href="/nlab/show/split+exact+sequence">split exact sequence</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+object">injective object</a>, <a class="existingWikiWord" href="/nlab/show/projective+object">projective object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+resolution">injective resolution</a>, <a class="existingWikiWord" href="/nlab/show/projective+resolution">projective resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/flat+resolution">flat resolution</a></p> </li> </ul> </li> </ul> <p><strong>Stable homotopy theory notions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a>, <a class="existingWikiWord" href="/nlab/show/enhanced+triangulated+category">enhanced triangulated category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+model+category">stable model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pretriangulated+dg-category">pretriangulated dg-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E-category">A-∞-category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+chain+complexes">(∞,1)-category of chain complexes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a>, <a class="existingWikiWord" href="/nlab/show/derived+functor+in+homological+algebra">derived functor in homological algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Tor">Tor</a>, <a class="existingWikiWord" href="/nlab/show/Ext">Ext</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lim%5E1+and+Milnor+sequences">lim^1 and Milnor sequences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fr-code">fr-code</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> </li> </ul> <p><strong>Constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/double+complex">double complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Koszul-Tate+resolution">Koszul-Tate resolution</a>, <a class="existingWikiWord" href="/nlab/show/BRST-BV+complex">BRST-BV complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+filtered+complex">spectral sequence of a filtered complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+double+complex">spectral sequence of a double complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+spectral+sequence">Grothendieck spectral sequence</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Leray+spectral+sequence">Leray spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Serre+spectral+sequence">Serre spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild-Serre+spectral+sequence">Hochschild-Serre spectral sequence</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong>Lemmas</strong></p> <p><a class="existingWikiWord" href="/nlab/show/diagram+chasing">diagram chasing</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/3x3+lemma">3x3 lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/four+lemma">four lemma</a>, <a class="existingWikiWord" href="/nlab/show/five+lemma">five lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/snake+lemma">snake lemma</a>, <a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/horseshoe+lemma">horseshoe lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Baer%27s+criterion">Baer's criterion</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/Schanuel%27s+lemma">Schanuel's lemma</a></p> <p><strong>Homology theories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cyclic+homology">cyclic homology</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> / <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal</a>, <a class="existingWikiWord" href="/nlab/show/operadic+Dold-Kan+correspondence">operadic</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Moore+complex">Moore complex</a>, <a class="existingWikiWord" href="/nlab/show/Alexander-Whitney+map">Alexander-Whitney map</a>, <a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+map">Eilenberg-Zilber map</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+theorem">Eilenberg-Zilber theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cochain+on+a+simplicial+set">cochain on a simplicial set</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#Statement'>Statement</a></li> <ul> <li><a href='#theorem_'>Theorem (<a class="existingWikiWord" href="/nlab/show/Tohoku">Tohoku</a>)</a></li> <li><a href='#theorem__2'>Theorem (<a class="existingWikiWord" href="/nlab/show/Tohoku">Tohoku</a>)</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <em>Grothendieck spectral sequence</em> is a <a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a> that computes the <a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a> of the <a class="existingWikiWord" href="/nlab/show/composition">composite</a> of two <a class="existingWikiWord" href="/nlab/show/derived+functor+in+homological+algebra">derived functors</a> on <a class="existingWikiWord" href="/nlab/show/categories+of+chain+complexes">categories of chain complexes</a>.</p> <h2 id="Statement">Statement</h2> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo>,</mo><mi>ℬ</mi><mo>,</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{A},\mathcal{B},\mathcal{C}</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian categories</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>𝒜</mi><mo>→</mo><mi>ℬ</mi></mrow><annotation encoding="application/x-tex">F \colon \mathcal{A}\to \mathcal{B}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo lspace="verythinmathspace">:</mo><mi>ℬ</mi><mo>→</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">G \colon \mathcal{B}\to \mathcal{C}</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/exact+functor">left exact</a> <a class="existingWikiWord" href="/nlab/show/additive+functor">additive</a> <a class="existingWikiWord" href="/nlab/show/functors">functors</a>. Assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo>,</mo><mi>ℬ</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}, \mathcal{B}</annotation></semantics></math> have <a class="existingWikiWord" href="/nlab/show/injective+object">enough injectives</a>.</p> <div class="num_theorem"> <h6 id="theorem_">Theorem (<a class="existingWikiWord" href="/nlab/show/Tohoku">Tohoku</a>)</h6> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>R</mi> <mi>F</mi></msub><mo>⊂</mo><mi mathvariant="normal">Ob</mi><mi>A</mi></mrow><annotation encoding="application/x-tex">R_F\subset \mathrm{Ob} A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>R</mi> <mi>G</mi></msub><mo>⊂</mo><mi mathvariant="normal">Ob</mi><mi>B</mi></mrow><annotation encoding="application/x-tex">R_G\subset\mathrm{Ob} B</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/class+of+adapted+objects">classes of objects adapted to</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> 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> respectively, and let furthermore <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mi>R</mi> <mi>A</mi></msub><mo stretchy="false">)</mo><mo>⊂</mo><msub><mi>R</mi> <mi>B</mi></msub></mrow><annotation encoding="application/x-tex">F(R_A)\subset R_B</annotation></semantics></math>. Then the derived functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>F</mi><mo>:</mo><msup><mi>D</mi> <mo>+</mo></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>D</mi> <mo>+</mo></msup><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R F:D^+(A)\to D^+(B)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>G</mi><mo>:</mo><msup><mi>D</mi> <mo>+</mo></msup><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>D</mi> <mo>+</mo></msup><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R G:D^+(B)\to D^+(C)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>G</mi><mo>∘</mo><mi>F</mi><mo stretchy="false">)</mo><mo>:</mo><msup><mi>D</mi> <mo>+</mo></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>D</mi> <mo>+</mo></msup><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R(G\circ F):D^+(A)\to D^+(C)</annotation></semantics></math> are defined and the natural morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>G</mi><mo>∘</mo><mi>F</mi><mo stretchy="false">)</mo><mo>→</mo><mi>R</mi><mi>G</mi><mo>∘</mo><mi>R</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">R(G\circ F)\to R G\circ R F</annotation></semantics></math> is an isomorphism.</p> </div> <div class="num_theorem"> <h6 id="theorem__2">Theorem (<a class="existingWikiWord" href="/nlab/show/Tohoku">Tohoku</a>)</h6> <p>In the above situation, assume that for every <a class="existingWikiWord" href="/nlab/show/injective+object">injective object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>∈</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">I \in \mathcal{A}</annotation></semantics></math> the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>ℬ</mi></mrow><annotation encoding="application/x-tex">F(I) \in \mathcal{B}</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/acyclic+object">acyclic object</a>.</p> <p>Then for every object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">A \in \mathcal{A}</annotation></semantics></math> there is a <a class="existingWikiWord" href="/nlab/show/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><mi>A</mi><mo stretchy="false">)</mo><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}(A)\}_{r,p,q}</annotation></semantics></math> called the <strong>Grothendieck spectral sequence</strong> whose <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 is the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>E</mi> <mn>2</mn> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</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><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> E^{p,q}_2(A) = R^p G \circ R^q F (A) </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/derived+functor+in+homological+algebra">right derived functors</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> in degrees <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math> and <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>, respectively and which is converging to to the derived functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>R</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>G</mi><mo>∘</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R^n(G\circ F)</annotation></semantics></math> of the composite 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>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>E</mi> <mn>∞</mn> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><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><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> E^{p,q}_\infty(A) \simeq R^{p+q}(G \circ F)(A) \,. </annotation></semantics></math></div> <p>Moreover, this is <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">A \in \mathcal{A}</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>By assumption of enough injectives, we may find an <em><a class="existingWikiWord" href="/nlab/show/injective+resolution">injective resolution</a></em></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>→</mo><mrow><msub><mo>≃</mo> <mi>qi</mi></msub></mrow></mover><msup><mi>C</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex"> A \stackrel{\simeq_{qi}}{\to} C^\bullet </annotation></semantics></math></div> <p>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>. Next, by the discussion at <em><a href="projective+resolution#ExistenceAndConstructionOfResolutionsOfComplexes">injective resolution – Existence and construction</a></em> we may find a <em>fully injective</em> resolution of the chain complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(C^\bullet)</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mi>I</mi> <mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msup><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> 0 \to F(C^\bullet) \to I^{\bullet, \bullet} \,, </annotation></semantics></math></div> <p>where hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>I</mi> <mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msup></mrow><annotation encoding="application/x-tex">I^{\bullet, \bullet}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/double+complex">double complex</a> of <a class="existingWikiWord" href="/nlab/show/injective+objects">injective objects</a> such that for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> the component <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mi>I</mi> <mrow><mi>n</mi><mo>,</mo><mo>•</mo></mrow></msup></mrow><annotation encoding="application/x-tex">0 \to F(C^n) \to I^{n,\bullet}</annotation></semantics></math> is an ordinary injective resolution of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>∈</mo><mi>ℬ</mi></mrow><annotation encoding="application/x-tex">F(C^n) \in \mathcal{B}</annotation></semantics></math>.</p> <p>Thus we have the corresponding double complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">(</mo><msup><mi>I</mi> <mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G(I^{\bullet,\bullet})</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>. The claim is that the Grothendieck spectral sequence is the <a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+double+complex">spectral sequence of a double complex</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">(</mo><msup><mi>I</mi> <mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G(I^{\bullet, \bullet})</annotation></semantics></math> equipped with the vertical-degree <a class="existingWikiWord" href="/nlab/show/filtered+chain+complex">filtration</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mrow></mrow> <mi>vert</mi></msup><msubsup><mi>E</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow> <mi>r</mi></msubsup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{{}^{vert}E^r_{p,q}(A)\}</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mrow></mrow> <mi>vert</mi></msup><msubsup><mi>E</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>R</mi> <mi>p</mi></msup><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"> {}^{vert} E^2_{p,q}(A) \simeq R^p G (R^q F(A)) \,. </annotation></semantics></math></div> <p>To see this, notice that by the assumption that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>I</mi> <mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msup></mrow><annotation encoding="application/x-tex">I^{\bullet,\bullet}</annotation></semantics></math> is a <em>fully</em> injective projective resolution, the short exact sequences</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><msup><mi>B</mi> <mrow><mi>q</mi><mo>,</mo><mi>p</mi></mrow></msup><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>Z</mi> <mrow><mi>q</mi><mo>,</mo><mi>p</mi></mrow></msup><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mrow><mi>q</mi><mo>,</mo><mi>p</mi></mrow></msup><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0 \to B^{q,p}(I) \to Z^{q,p}(I) \to H^{q,p}(I) \to 0 </annotation></semantics></math></div> <p>are <a class="existingWikiWord" href="/nlab/show/split+exact+sequence">split</a> (by the discussion there) and hence so is their image under any functor and hence in particular under <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>. Accordingly we have</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><msup><mrow></mrow> <mi>vert</mi></msup><msubsup><mi>E</mi> <mn>1</mn> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup></mtd> <mtd><mo>≃</mo><msup><mi>H</mi> <mi>q</mi></msup><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">(</mo><msup><mi>I</mi> <mrow><mo>•</mo><mo>,</mo><mi>p</mi></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">(</mo><msup><mi>Z</mi> <mrow><mi>q</mi><mo>,</mo><mi>p</mi></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">(</mo><msup><mi>B</mi> <mrow><mi>q</mi><mo>,</mo><mi>p</mi></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>G</mi><msup><mi>H</mi> <mrow><mi>q</mi><mo>,</mo><mi>p</mi></mrow></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} {}^{vert}E^{p,q}_1 & \simeq H^q(G(I^{\bullet,p})) \\ & \simeq (G(Z^{q,p})) / (G(B^{q,p})) \\ & \simeq G H^{q,p} \end{aligned} </annotation></semantics></math></div> <p>(the first two equivalences by general properties of the filtration spectral sequence, the last by the above splitness). Hence it follows that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msup><mrow></mrow> <mi>vert</mi></msup><msubsup><mi>E</mi> <mn>2</mn> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup></mtd> <mtd><mo>≃</mo><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">(</mo><msup><mi>H</mi> <mrow><mi>q</mi><mo>,</mo><mo>•</mo></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msup><mi>R</mi> <mi>p</mi></msup><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></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} {}^{vert}E^{p,q}_2 & \simeq H^p(G(H^{q,\bullet})) \\ & \simeq R^p G (R^q F (A)) \end{aligned} \,, </annotation></semantics></math></div> <p>where in the last step we used that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mrow><mi>q</mi><mo>,</mo><mo>•</mo></mrow></msup></mrow><annotation encoding="application/x-tex">H^{q,\bullet}</annotation></semantics></math> is be construction an injective resolution of <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><msup><mi>C</mi> <mo>•</mo></msup><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(C^\bullet)) \simeq R^q F(A)</annotation></semantics></math> (using the <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>-acyclicity of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(C^\bullet)</annotation></semantics></math>).</p> <p>This establishes the spectral sequence and its second page as claimed. It remains to determine its convergence.</p> <p>To that end, consider dually, the spectral sequence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mrow></mrow> <mi>hor</mi></msup><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">\{{}^{hor}E^{p,q}_r\}</annotation></semantics></math> coming from the horizontal filtration on the double complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">(</mo><msup><mi>I</mi> <mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G(I^{\bullet, \bullet})</annotation></semantics></math>. By the general properties of <a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+double+complex">spectral sequence of a double complex</a> this converges to the same value as the previous one. But for this latter spectral sequence we find</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><msup><mrow></mrow> <mi>hor</mi></msup><msubsup><mi>E</mi> <mn>1</mn> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup></mtd> <mtd><mo>≃</mo><msup><mi>H</mi> <mi>q</mi></msup><mo stretchy="false">(</mo><mi>G</mi><msup><mi>I</mi> <mrow><mi>p</mi><mo>,</mo><mo>•</mo></mrow></msup><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msup><mi>R</mi> <mi>q</mi></msup><mi>G</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mi>p</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} {}^{hor}E^{p,q}_1 & \simeq H^q(G I^{p,\bullet}) \\ & \simeq R^q G(F(C^p)) \end{aligned} \,, </annotation></semantics></math></div> <p>the first equivalence by the general properties of filtration spectral sequences, the second then by the definition of <a class="existingWikiWord" href="/nlab/show/derived+functor+in+homological+algebra">right derived functors</a>. But by assumption <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mi>p</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(C^p)</annotation></semantics></math> is <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+object">acyclic</a> and hence all these derived functors vanish in positive degree, so that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mrow></mrow> <mi>hor</mi></msup><msubsup><mi>E</mi> <mn>1</mn> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup><mo>≃</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>G</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mi>p</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mi>if</mi><mspace width="thickmathspace"></mspace><mi>q</mi><mo>=</mo><mn>0</mn></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mi>otherwise</mi></mtd></mtr></mtable></mrow></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> {}^{hor}E^{p,q}_1 \simeq \left\{ \array{ G(F(C^p)) & if\; q = 0 \\ 0 & otherwise } \right. \,. </annotation></semantics></math></div> <p>Next, 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 then contains just horizontal homology of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G(F(C^\bullet))</annotation></semantics></math> and this is by definition now the derived functor of the composite 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> with <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>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mrow></mrow> <mi>hor</mi></msup><msubsup><mi>E</mi> <mn>2</mn> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup><mo>≃</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><msup><mi>R</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mi>G</mi><mo>∘</mo><mi>F</mi><mo stretchy="false">)</mo></mtd> <mtd><mi>if</mi><mspace width="thickmathspace"></mspace><mi>q</mi><mo>=</mo><mn>0</mn></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mi>otherwise</mi></mtd></mtr></mtable></mrow></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> {}^{hor}E^{p,q}_2 \simeq \left\{ \array{ R^p(G \circ F) & if \; q = 0 \\ 0 & otherwise } \right. \,. </annotation></semantics></math></div> <p>Since this is concentrated in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>q</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(q = 0)</annotation></semantics></math>-row the spectral sequence of the horizontal filtration collapses here and hence</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><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>Tot</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">(</mo><msup><mi>I</mi> <mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><msup><mi>G</mi> <mi>n</mi></msup><msup><mi>H</mi> <mrow><mi>n</mi><mo>+</mo><mn>0</mn></mrow></msup><mo stretchy="false">(</mo><mi>Tot</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">(</mo><msup><mi>I</mi> <mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msubsup><mi>E</mi> <mn>∞</mn> <mrow><mi>n</mi><mo>,</mo><mn>0</mn></mrow></msubsup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} H^n(Tot(G(I^{\bullet,\bullet}))) &\simeq G^n H^{n+0}(Tot(G(I^{\bullet,\bullet}))) \\ & \simeq E^{n,0}_\infty \end{aligned} </annotation></semantics></math></div> <p>So in conclusion we have</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><msup><mi>R</mi> <mi>p</mi></msup><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></mtd> <mtd><mo>≃</mo><msup><mrow></mrow> <mi>vert</mi></msup><msubsup><mi>E</mi> <mn>2</mn> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>⇒</mo><msup><mrow></mrow> <mi>vert</mi></msup><msubsup><mi>E</mi> <mn>∞</mn> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msubsup><mi>G</mi> <mi>vert</mi> <mi>p</mi></msubsup><msup><mi>H</mi> <mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></msup><mo stretchy="false">(</mo><mi>Tot</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">(</mo><msup><mi>I</mi> <mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msup><mi>H</mi> <mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></msup><mo stretchy="false">(</mo><mi>Tot</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">(</mo><msup><mi>I</mi> <mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msubsup><mi>G</mi> <mi>hor</mi> <mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></msubsup><msup><mi>H</mi> <mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></msup><mo stretchy="false">(</mo><mi>Tot</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">(</mo><msup><mi>I</mi> <mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msup><mrow></mrow> <mi>hor</mi></msup><msubsup><mi>E</mi> <mn>∞</mn> <mrow><mi>p</mi><mo>+</mo><mi>q</mi><mo>,</mo><mn>0</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><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></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} R^p G(R^q F(A)) & \simeq {}^{vert}E^{p,q}_2 \\ & \Rightarrow {}^{vert} E^{p,q}_\infty \\ & \simeq G^p_{vert} H^{p+q}(Tot(G(I^{\bullet, \bullet}))) \\ & \simeq H^{p+q}(Tot(G(I^{\bullet, \bullet}))) \\ & \simeq G^{p+q}_{hor} H^{p+q}(Tot(G(I^{\bullet, \bullet}))) \\ & \simeq {}^{hor} E^{p+q,0}_\infty(A) \\ & \simeq R^{p+q}(G \circ F)(A) \end{aligned} </annotation></semantics></math></div></div> <h2 id="examples">Examples</h2> <p>Many other classes of spectral sequences are special cases of the Grothendieck spectral sequence, for instance the</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild-Serre+spectral+sequence">Hochschild-Serre spectral sequence</a></p> </li> <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/base+change+spectral+sequence">base change spectral sequence</a></p> </li> </ul> <h2 id="references">References</h2> <p>Leture notes include</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/James+Milne">James Milne</a>, section 10 of <em><a class="existingWikiWord" href="/nlab/show/Lectures+on+%C3%89tale+Cohomology">Lectures on Étale Cohomology</a></em></p> </li> <li> <p>Jinhyun Park, <em>Personal notes on Grothendieck spectral sequence</em> (<a href="http://mathsci.kaist.ac.kr/~jinhyun/note/g_s_sequence/sequence.pdf">pdf</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on April 22, 2020 at 13:17:00. See the <a href="/nlab/history/Grothendieck+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/Grothendieck+spectral+sequence" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/4450/#Item_1">Discuss</a><span class="backintime"><a href="/nlab/revision/Grothendieck+spectral+sequence/11" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/Grothendieck+spectral+sequence" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/Grothendieck+spectral+sequence" accesskey="S" class="navlink" id="history" rel="nofollow">History (11 revisions)</a> <a href="/nlab/show/Grothendieck+spectral+sequence/cite" style="color: black">Cite</a> <a href="/nlab/print/Grothendieck+spectral+sequence" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/Grothendieck+spectral+sequence" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>