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> pure motive 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> pure motive </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/14244/#Item_3" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="motivic_cohomology">Motivic cohomology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/motive">motive</a></strong>, <strong><a class="existingWikiWord" href="/nlab/show/motivic+cohomology">motivic cohomology</a></strong></p> <h2 id="ingredients">Ingredients</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a></p> </li> </ul> <h2 id="definitions">Definitions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pure+motive">pure motive</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chow+motive">Chow motive</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/numerical+motive">numerical motive</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/periods">periods</a>, <a class="existingWikiWord" href="/nlab/show/motivic+Galois+group">motivic Galois group</a>, <a class="existingWikiWord" href="/nlab/show/cosmic+Galois+group">cosmic Galois group</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mixed+motive">mixed motive</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Nori+motive">Nori motive</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Voevodsky+motive">Voevodsky motive</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A1-homotopy+theory">A1-homotopy theory</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/noncommutative+motive">noncommutative motive</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/bivariant+algebraic+K-theory">bivariant algebraic K-theory</a>, <a class="existingWikiWord" href="/nlab/show/KK-theory">KK-theory</a></li> </ul> </li> </ul> <h2 id="related">Related</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/motivic+function">motivic function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivic+integration">motivic integration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivic+Donaldson-Thomas+invariant">motivic Donaldson-Thomas invariant</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/motivic+cohomology+-+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='#construction'>Construction</a></li> <ul> <li><a href='#category_of_correspondences'>Category of correspondences</a></li> <li><a href='#category_of_effective_pure_motives'>Category of effective pure motives</a></li> <li><a href='#category_of_pure_motives'>Category of pure motives</a></li> <li><a href='#category_of_pure_chow_motives'>Category of pure Chow motives</a></li> <li><a href='#category_of_pure_numerical_motives'>Category of pure numerical motives</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck">Grothendieck</a><a class="existingWikiWord" href="/nlab/show/conjecture">conjectured</a> that every <a class="existingWikiWord" href="/nlab/show/Weil+cohomology+theory">Weil cohomology theory</a> factors uniquely through some <a class="existingWikiWord" href="/nlab/show/category">category</a>, which he called the category of <em><a class="existingWikiWord" href="/nlab/show/motives">motives</a></em>. For <a class="existingWikiWord" href="/nlab/show/smooth+variety">smooth</a> <a class="existingWikiWord" href="/nlab/show/projective+varieties">projective varieties</a> (over some <a class="existingWikiWord" href="/nlab/show/field">field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>) such a category was given by Grothendieck himself, called the category of <em>pure <a class="existingWikiWord" href="/nlab/show/Chow+motives">Chow motives</a></em>. For general smooth varieties the category is still conjectural, see at <em><a class="existingWikiWord" href="/nlab/show/mixed+motives">mixed motives</a></em>.</p> <h2 id="construction">Construction</h2> <p>Fix some <a class="existingWikiWord" href="/nlab/show/adequate+equivalence+relation">adequate equivalence relation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∼</mo></mrow><annotation encoding="application/x-tex">\sim</annotation></semantics></math> (e.g. <a class="existingWikiWord" href="/nlab/show/rational+equivalence">rational equivalence</a>). Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Z</mi> <mi>i</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z^i(X)</annotation></semantics></math> denote the group of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/codimension">codimensional</a> <a class="existingWikiWord" href="/nlab/show/algebraic+cycles">algebraic cycles</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>A</mi> <mo>∼</mo> <mi>i</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A^i_\sim(X)</annotation></semantics></math> denote the <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Z</mi> <mi>i</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo>∼</mo></mrow><annotation encoding="application/x-tex">Z^i(X)/\sim</annotation></semantics></math>.</p> <h3 id="category_of_correspondences">Category of correspondences</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Corr</mi> <mo>∼</mo></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Corr_\sim(k)</annotation></semantics></math>, the category of <a class="existingWikiWord" href="/nlab/show/correspondences">correspondences</a>, be the <a class="existingWikiWord" href="/nlab/show/category">category</a> whose <a class="existingWikiWord" href="/nlab/show/objects">objects</a> are <a class="existingWikiWord" href="/nlab/show/smooth+variety">smooth</a> <a class="existingWikiWord" href="/nlab/show/projective+varieties">projective varieties</a> and whose <a class="existingWikiWord" href="/nlab/show/hom-sets">hom-sets</a> are 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><msub><mi>Corr</mi> <mo>∼</mo></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>h</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">⨁</mo> <mi>i</mi></munder><msubsup><mi>A</mi> <mo>∼</mo> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>i</mi></msub><mo>×</mo><mi>Y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> Corr_\sim(h(X),h(Y)) = \bigoplus_i A^{n_i}_\sim(X_i \times Y) \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X_i)</annotation></semantics></math> are the <a class="existingWikiWord" href="/nlab/show/irreducible+components">irreducible components</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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">n_i</annotation></semantics></math> are their respective <a class="existingWikiWord" href="/nlab/show/dimensions">dimensions</a>. The <a class="existingWikiWord" href="/nlab/show/composition">composition</a> of two <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>∈</mo><mi>Corr</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\alpha \in Corr(X,Y)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi><mo>∈</mo><mi>Corr</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\beta \in Corr(Y,Z)</annotation></semantics></math> is given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mrow><mi>XZ</mi><mo>,</mo><mo>*</mo></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>p</mi> <mi>XY</mi> <mo>*</mo></msubsup><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo><mo>.</mo><msubsup><mi>p</mi> <mi>YZ</mi> <mo>*</mo></msubsup><mo stretchy="false">(</mo><mi>β</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> p_{XZ,*} (p_{XY}^*(\alpha) . p_{YZ}^*(\beta)) </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>XY</mi></msub></mrow><annotation encoding="application/x-tex">p_{XY}</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/projection">projection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>Y</mi><mo>×</mo><mi>Z</mi><mo>→</mo><mi>X</mi><mo>×</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \times Y \times Z \to X \times Y</annotation></semantics></math> and so on, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>.</mo></mrow><annotation encoding="application/x-tex">.</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/intersection+product">intersection product</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>Y</mi><mo>×</mo><mi>Z</mi></mrow><annotation encoding="application/x-tex">X \times Y \times Z</annotation></semantics></math>.</p> <p>There is a canonical <a class="existingWikiWord" href="/nlab/show/contravariant+functor">contravariant functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo lspace="verythinmathspace">:</mo><mi>SmProj</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>Corr</mi> <mo>∼</mo></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> h \colon SmProj(k) \to Corr_\sim(k) </annotation></semantics></math></div> <p>from the category of smooth projective varieties over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> given by <a class="existingWikiWord" href="/nlab/show/mapping">mapping</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>↦</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X \mapsto X</annotation></semantics></math> and a morphism <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> to its <a class="existingWikiWord" href="/nlab/show/graph+of+a+function">graph</a>, the <a class="existingWikiWord" href="/nlab/show/image">image</a> of its <a class="existingWikiWord" href="/nlab/show/graph+morphism">graph morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>f</mi></msub><mo>:</mo><mi>X</mi><mo>→</mo><mi>X</mi><mo>×</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">\Gamma_f : X \to X \times Y</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/category+of+correspondences">category of correspondences</a> is <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>h</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>h</mi><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h(X) \otimes h(Y) \coloneqq h(X \times Y)</annotation></semantics></math>.</p> <p>We also define a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Corr</mi> <mo>∼</mo></msub><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Corr_\sim(k, R)</annotation></semantics></math> of correspondences with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in some <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> <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>, by <a class="existingWikiWord" href="/nlab/show/tensor+product">tensoring</a> the morphisms with <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>; this is a <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/linear+category">linear category</a> <a class="existingWikiWord" href="/nlab/show/additive+category">additive</a> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal</a> category.</p> <h3 id="category_of_effective_pure_motives">Category of effective pure motives</h3> <p>The <a class="existingWikiWord" href="/nlab/show/Karoubi+envelope">Karoubi envelope</a> (pseudo-abelianisation) of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Corr</mi> <mo>∼</mo></msub><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Corr_\sim(k, R)</annotation></semantics></math> is called the category of <strong>effective pure motives</strong> (with coefficients 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> and with respect to the equivalence relation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∼</mo></mrow><annotation encoding="application/x-tex">\sim</annotation></semantics></math>), denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Mot</mi> <mo>∼</mo> <mi>eff</mi></msubsup><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Mot^eff_\sim(k, R)</annotation></semantics></math>.</p> <p>Explicitly its objects are <a class="existingWikiWord" href="/nlab/show/pairs">pairs</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(h(X), p)</annotation></semantics></math> with <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 smooth projective variety and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∈</mo><mi>Corr</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>h</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p \in Corr(h(X), h(X))</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/idempotent">idempotent</a>, and <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(h(X), p)</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(h(Y), q)</annotation></semantics></math> are morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>h</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h(X) \to h(Y)</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Corr</mi> <mo>∼</mo></msub></mrow><annotation encoding="application/x-tex">Corr_\sim</annotation></semantics></math> of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>∘</mo><mi>α</mi><mo>∘</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">q \circ \alpha \circ p</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>∈</mo><msub><mi>Corr</mi> <mo>∼</mo></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>h</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\alpha \in Corr_{\sim}(h(X), h(Y))</annotation></semantics></math>.</p> <p>This is still a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal</a> category with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo>⊗</mo><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>,</mo><mi>p</mi><mo>×</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(h(X), p) \otimes (h(Y), q) = (h(X \times Y), p \times q)</annotation></semantics></math>. Further it is a <a class="existingWikiWord" href="/nlab/show/Karoubian+category">Karoubian</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/linear+category">linear</a> and <a class="existingWikiWord" href="/nlab/show/additive+category">additive</a>.</p> <p>The image of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>SmProj</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \in SmProj(k)</annotation></semantics></math> under the above functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo lspace="verythinmathspace">:</mo><mi>SmProj</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>Corr</mi> <mo>∼</mo></msub><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><msubsup><mi>Mot</mi> <mo>∼</mo> <mi>eff</mi></msubsup><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> h \colon SmProj(k) \to Corr_\sim(k,A) \to Mot^{eff}_\sim(k,R) </annotation></semantics></math></div> <p>is the <strong>the motive 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></strong>.</p> <h3 id="category_of_pure_motives">Category of pure motives</h3> <p>There exists a motive <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>L</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{L}</annotation></semantics></math>, called the <strong><a class="existingWikiWord" href="/nlab/show/Lefschetz+motive">Lefschetz motive</a></strong>, such that the motive of the <a class="existingWikiWord" href="/nlab/show/projective+line">projective line</a> decomposes as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><msubsup><mstyle mathvariant="bold"><mi>P</mi></mstyle> <mi>k</mi> <mn>1</mn></msubsup><mo stretchy="false">)</mo><mo>=</mo><mi>h</mi><mo stretchy="false">(</mo><mo lspace="0em" rspace="thinmathspace">Spec</mo><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⊕</mo><mstyle mathvariant="bold"><mi>L</mi></mstyle></mrow><annotation encoding="application/x-tex">h(\mathbf{P}^1_k) = h(\Spec(k)) \oplus \mathbf{L}</annotation></semantics></math></div> <p>To get a <a class="existingWikiWord" href="/nlab/show/rigid+category">rigid category</a> we formally invert the Lefschetz motive and get a category</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Mot</mi> <mo>∼</mo></msub><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>R</mi><mo stretchy="false">)</mo><mo>≔</mo><msubsup><mi>Mot</mi> <mo>∼</mo> <mi>eff</mi></msubsup><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>R</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><msup><mstyle mathvariant="bold"><mi>L</mi></mstyle> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> Mot_\sim(k, R) \coloneqq Mot^{eff}_\sim(k,R)[\mathbf{L}^{-1}] \,, </annotation></semantics></math></div> <p>the <strong>category of pure motives</strong> (with coefficients 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> and with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∼</mo></mrow><annotation encoding="application/x-tex">\sim</annotation></semantics></math>).</p> <p>This is a <a class="existingWikiWord" href="/nlab/show/rigid+category">rigid</a>, <a class="existingWikiWord" href="/nlab/show/Karoubian+category">Karoubian</a>, <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a>. Its objects are triples <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>p</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(h(X), p, n)</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>Z</mi></mstyle></mrow><annotation encoding="application/x-tex">n \in \mathbf{Z}</annotation></semantics></math>.</p> <h3 id="category_of_pure_chow_motives">Category of pure Chow motives</h3> <p>When the relation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∼</mo></mrow><annotation encoding="application/x-tex">\sim</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/rational+equivalence">rational equivalence</a> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>A</mi> <mo>∼</mo> <mo>*</mo></msubsup></mrow><annotation encoding="application/x-tex">A^*_\sim</annotation></semantics></math> are the <a class="existingWikiWord" href="/nlab/show/Chow+groups">Chow groups</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mot</mi> <mo>∼</mo></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Mot</mi> <mi>rat</mi></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Mot_\sim(k) = Mot_{rat}(k)</annotation></semantics></math> is called the category of <strong>pure <a class="existingWikiWord" href="/nlab/show/Chow+motives">Chow motives</a></strong>. This category has the advantage that it is universal for <a class="existingWikiWord" href="/nlab/show/Weil+cohomology+theory">Weil cohomology theories</a>: that is, every Weil cohomology factors uniquely through it.</p> <h3 id="category_of_pure_numerical_motives">Category of pure numerical motives</h3> <p>When the relation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∼</mo></mrow><annotation encoding="application/x-tex">\sim</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/numerical+equivalence">numerical equivalence</a>, then one obtains <em><a class="existingWikiWord" href="/nlab/show/numerical+motives">numerical motives</a></em>. This category has the advantage of being a <a class="existingWikiWord" href="/nlab/show/semisimple+category">semisimple</a> <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>. In fact, <a class="existingWikiWord" href="/nlab/show/Uwe+Jannsen">Uwe Jannsen</a> proved that numerical equivalence is the only <a class="existingWikiWord" href="/nlab/show/adequate+equivalence+relation">adequate equivalence relation</a> that gives a semisimple abelian category of pure motives.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+with+transfer">sheaf with transfer</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motive">motive</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mixed+motive">mixed motive</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Voevodsky+motive">Voevodsky motive</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/noncommutative+motive">noncommutative motive</a></p> </li> </ul> <h2 id="references">References</h2> <ul id="Sujatha"> <li> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Dugger">Daniel Dugger</a>, <em>Navigating the Motivic World</em>. (<a href="http://www.stat.ucla.edu/~ywu/wbook.pdf">pdf</a>) A draft of a user-friendly introduction to motives, especially good if you’re coming to this topic through algebraic topology.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Uwe+Jannsen">Uwe Jannsen</a>, Motives, numerical equivalence, and semi-simplicity, Invent. Math. 107.3 (1992): 447-452. (<a href="https://epub.uni-regensburg.de/26642/1/jannsen10.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Minhyong+Kim">Minhyong Kim</a>, <em>Classical Motives: Motivic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math>-functions</em> (<a href="http://www.ucl.ac.uk/~ucahmki/ihes3.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bruno+Kahn">Bruno Kahn</a>, <a href="http://www.aimath.org/WWN/motivesdessins/PaloAlto1.pdf">pdf slides</a> on pure motives</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yuri+Manin">Yuri Manin</a>, <em>Correspondences, motifs and monoidal transformations</em> , Math. USSR Sb. 6 439, 1968(<a href="http://resources.agssp2012.torsor.org/documents/manin.pdf">pdf</a>, <a href="http://iopscience.iop.org/0025-5734/6/4/A01">web</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/James+Milne">James Milne</a>, <em>Motives – Grothendieck’s Dream</em> (<a href="http://www.jmilne.org/math/xnotes/MOT.pdf">pdf</a>)</p> </li> <li> <p><span class="newWikiWord">Tony Scholl<a href="/nlab/new/Tony+Scholl">?</a></span>, <em>Classical motives</em>, in Motives, Seattle 1991. Proc Symp. Pure Math 55 (1994), part 1, 163-187 (<a href="https://www.dpmms.cam.ac.uk/~ajs1005/preprints/classical_motives.pdf">pdf</a>)</p> </li> <li> <p>R. Sujatha, <em>Motives from a categorical point of view</em>, Lecture notes (2008) (<a href="http://www.math.tifr.res.in/~sujatha/ihes.pdf">pdf</a>)</p> </li> </ul> <p>Section 8.2 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Alain+Connes">Alain Connes</a>, <a class="existingWikiWord" href="/nlab/show/Matilde+Marcolli">Matilde Marcolli</a>, <em><a class="existingWikiWord" href="/nlab/show/Noncommutative+Geometry%2C+Quantum+Fields+and+Motives">Noncommutative Geometry, Quantum Fields and Motives</a></em> (<a href="http://www.alainconnes.org/docs/bookwebfinal.pdf">pdf</a>)</li> </ul> <div class="property">category: <a class="category_link" href="/nlab/all_pages/algebraic+geometry">algebraic geometry</a></div></body></html> </div> <div class="revisedby"> <p> Last revised on July 3, 2023 at 05:36:24. See the <a href="/nlab/history/pure+motive" 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/pure+motive" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/14244/#Item_3">Discuss</a><span class="backintime"><a href="/nlab/revision/pure+motive/29" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/pure+motive" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/pure+motive" accesskey="S" class="navlink" id="history" rel="nofollow">History (29 revisions)</a> <a href="/nlab/show/pure+motive/cite" style="color: black">Cite</a> <a href="/nlab/print/pure+motive" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/pure+motive" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>