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> Picard group 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> Picard group </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/3005/#Item_10" 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="monoidal_categories">Monoidal categories</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+monoidal+category">enriched monoidal category</a>, <a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+diagram">string diagram</a>, <a class="existingWikiWord" href="/nlab/show/tensor+network">tensor network</a></p> </li> </ul> <p><strong>With braiding</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/balanced+monoidal+category">balanced monoidal category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twist">twist</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a></p> </li> </ul> <p><strong>With duals for objects</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+with+duals">category with duals</a> (list of them)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dualizable+object">dualizable object</a> (what they have)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid monoidal category</a>, a.k.a. <a class="existingWikiWord" href="/nlab/show/autonomous+category">autonomous category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pivotal+category">pivotal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spherical+category">spherical category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ribbon+category">ribbon category</a>, a.k.a. <a class="existingWikiWord" href="/nlab/show/tortile+category">tortile category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+closed+category">compact closed category</a></p> </li> </ul> <p><strong>With duals for morphisms</strong></p> <ul> <li> <p><span class="newWikiWord">monoidal dagger-category<a href="/nlab/new/monoidal+dagger-category">?</a></span></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+dagger-category">symmetric monoidal dagger-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dagger+compact+category">dagger compact category</a></p> </li> </ul> <p><strong>With traces</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/trace">trace</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/traced+monoidal+category">traced monoidal category</a></p> </li> </ul> <p><strong>Closed structure</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closed category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+category">closed category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/star-autonomous+category">star-autonomous category</a></p> </li> </ul> <p><strong>Special sorts of products</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">cartesian monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semicartesian+monoidal+category">semicartesian monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+category+with+diagonals">monoidal category with diagonals</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multicategory">multicategory</a></p> </li> </ul> <p><strong>Semisimplicity</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/semisimple+category">semisimple category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fusion+category">fusion category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/modular+tensor+category">modular tensor category</a></p> </li> </ul> <p><strong>Morphisms</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+functor">monoidal functor</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/lax+monoidal+functor">lax</a>, <a class="existingWikiWord" href="/nlab/show/oplax+monoidal+functor">oplax</a>, <a class="existingWikiWord" href="/nlab/show/strong+monoidal+functor">strong</a> <a class="existingWikiWord" href="/nlab/show/bilax+monoidal+functor">bilax</a>, <a class="existingWikiWord" href="/nlab/show/Frobenius+monoidal+functor">Frobenius</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/braided+monoidal+functor">braided monoidal functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+functor">symmetric monoidal functor</a></p> </li> </ul> <p><strong>Internal monoids</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+in+a+monoidal+category">monoid in a monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+a+symmetric+monoidal+category">commutative monoid in a symmetric monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module+over+a+monoid">module over a monoid</a></p> </li> </ul> <p><strong id="_examples">Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+monoidal+structure+on+presheaves">closed monoidal structure on presheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Day+convolution">Day convolution</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/coherence+theorem+for+monoidal+categories">coherence theorem for monoidal categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> </ul> <p><strong>In higher category theory</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+2-category">monoidal 2-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/braided+monoidal+2-category">braided monoidal 2-category</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+bicategory">monoidal bicategory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cartesian+bicategory">cartesian bicategory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/k-tuply+monoidal+n-category">k-tuply monoidal n-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/little+cubes+operad">little cubes operad</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+double+category">compact double category</a></p> </li> </ul> </div></div> <h4 id="group_theory">Group Theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/group+theory">group theory</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></li> <li><a class="existingWikiWord" href="/nlab/show/group+object">group object</a>, <a class="existingWikiWord" href="/nlab/show/group+object+in+an+%28%E2%88%9E%2C1%29-category">group object in an (∞,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>, <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></li> <li><a class="existingWikiWord" href="/nlab/show/super+abelian+group">super abelian group</a></li> <li><a class="existingWikiWord" href="/nlab/show/group+action">group action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></li> <li><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></li> <li><a class="existingWikiWord" href="/nlab/show/progroup">progroup</a></li> <li><a class="existingWikiWord" href="/nlab/show/homogeneous+space">homogeneous space</a></li> </ul> <p><strong>Classical groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a>. <a class="existingWikiWord" href="/nlab/show/projective+unitary+group">projective unitary group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+group">symplectic group</a></p> </li> </ul> <p><strong>Finite groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/finite+group">finite group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+group">symmetric group</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a>, <a class="existingWikiWord" href="/nlab/show/braid+group">braid group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classification+of+finite+simple+groups">classification of finite simple groups</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sporadic+finite+simple+groups">sporadic finite simple groups</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Monster+group">Monster group</a>, <a class="existingWikiWord" href="/nlab/show/Mathieu+group">Mathieu group</a></li> </ul> </li> </ul> <p><strong>Group schemes</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+variety">abelian variety</a></p> </li> </ul> <p><strong>Topological groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+topological+group">compact topological group</a>, <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+group">locally compact topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/maximal+compact+subgroup">maximal compact subgroup</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+group">string group</a></p> </li> </ul> <p><strong>Lie groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact Lie group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kac-Moody+group">Kac-Moody group</a></p> </li> </ul> <p><strong>Super-Lie groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Lie+group">super Lie group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Euclidean+group">super Euclidean group</a></p> </li> </ul> <p><strong>Higher groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-group">2-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/crossed+module">crossed module</a>, <a class="existingWikiWord" href="/nlab/show/strict+2-group">strict 2-group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-group">n-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crossed+complex">crossed complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/k-tuply+groupal+n-groupoid">k-tuply groupal n-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle+n-group">circle n-group</a>, <a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a>, <a class="existingWikiWord" href="/nlab/show/fivebrane+Lie+6-group">fivebrane Lie 6-group</a></p> </li> </ul> <p><strong>Cohomology and Extensions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a>, <a class="existingWikiWord" href="/nlab/show/Ext-group">Ext-group</a></p> </li> </ul> <p><strong>Related concepts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/quantum+group">quantum group</a></li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#general_abstract'>General abstract</a></li> <li><a href='#in_algebraic_geometry'>In algebraic geometry</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#properties'>Properties</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#in_algebraic_geometry_2'>In algebraic geometry</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>Fully generally, a <em>Picard group</em> is an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> defined for a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> as the group of <a class="existingWikiWord" href="/nlab/show/isomorphism+classes">isomorphism classes</a> of <a class="existingWikiWord" href="/nlab/show/objects">objects</a> which are <a class="existingWikiWord" href="/nlab/show/invertible+object">invertible</a> with respect to the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a>.</p> <p>Traditionally though one speaks in the context of <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a> of the <em>Picard group</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Pic</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Pic(X)</annotation></semantics></math> of some kind of <a class="existingWikiWord" href="/nlab/show/space">space</a> and by default means the invertible objects in some monoidal category of something like <a class="existingWikiWord" href="/nlab/show/vector+bundles">vector bundles</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. Specifically for <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 <a class="existingWikiWord" href="/nlab/show/ringed+topos">ringed topos</a> (in particular a <a class="existingWikiWord" href="/nlab/show/ringed+space">ringed space</a>), then the <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> to be understood is that of locally free <a class="existingWikiWord" href="/nlab/show/module+sheaves">module sheaves</a> over the <a class="existingWikiWord" href="/nlab/show/structure+sheaf">structure sheaf</a> and hence the Picard group in this case is that of locally free sheaves of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{O}_X</annotation></semantics></math>-modules of <a class="existingWikiWord" href="/nlab/show/rank">rank</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math> (i.e. the <a class="existingWikiWord" href="/nlab/show/line+bundles">line bundles</a>).</p> <p>Specifically in <a class="existingWikiWord" href="/nlab/show/complex+geometry">complex geometry</a> these objects on a <a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/holomorphic+vector+bundles">holomorphic vector bundles</a> and hence in this case the Picard group of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is that of isomorphism classes of <a class="existingWikiWord" href="/nlab/show/holomorphic+line+bundles">holomorphic line bundles</a>. This case has an obvious genralization to <a class="existingWikiWord" href="/nlab/show/schemes">schemes</a> in <a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a>, and in much of the literature a <em>Picard group</em> is meant to be a Picard group of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔾</mi> <mi>m</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{G}_m</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/torsors">torsors</a> over a given <a class="existingWikiWord" href="/nlab/show/scheme">scheme</a>. In this (and other) geometric situations, the Picard group naturally inherits geometric structure itself and equipped with that it is then called the <em><a class="existingWikiWord" href="/nlab/show/Picard+scheme">Picard scheme</a></em> (with <a class="existingWikiWord" href="/nlab/show/formal+geometry">formal</a> completion the <em><a class="existingWikiWord" href="/nlab/show/formal+Picard+group">formal Picard group</a></em>), see there for more.</p> <p>Not <a class="existingWikiWord" href="/nlab/show/decategorification">decategorifying</a> by passing to isomorphism classes instead yields the concept of <a class="existingWikiWord" href="/nlab/show/Picard+2-group">Picard 2-group</a> and geometrically that of <a class="existingWikiWord" href="/nlab/show/Picard+stack">Picard stack</a>, see there for more.</p> <h2 id="definition">Definition</h2> <h3 id="general_abstract">General abstract</h3> <div class="num_definition"> <h6 id="definition_2">Definition</h6> <p>Given a (<a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a>) <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mo>⊗</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C, \otimes)</annotation></semantics></math>, the <em>Picard group of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mo>⊗</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C,\otimes)</annotation></semantics></math> is the group of <a class="existingWikiWord" href="/nlab/show/isomorphism+classes">isomorphism classes</a> of <a class="existingWikiWord" href="/nlab/show/invertible+objects">invertible objects</a>, those that have an <a class="existingWikiWord" href="/nlab/show/inverse">inverse</a> under the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> – the <a class="existingWikiWord" href="/nlab/show/line+objects">line objects</a>. Equivalently, this is the <a class="existingWikiWord" href="/nlab/show/decategorification">decategorification</a> or <a class="existingWikiWord" href="/nlab/show/0-truncated">0-truncation</a> of the <a class="existingWikiWord" href="/nlab/show/Picard+2-group">Picard 2-group</a>, the maximal <a class="existingWikiWord" href="/nlab/show/2-group">2-group</a> inside a monoidal category.</em></p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>The Picard group is indeed a <a class="existingWikiWord" href="/nlab/show/group">group</a>: First, if <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">\mathcal{L}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℳ</mi></mrow><annotation encoding="application/x-tex">\mathcal{M}</annotation></semantics></math> are elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Pic</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Pic(X)</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mo>⊗</mo><mi>ℳ</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}\otimes \mathcal{M}</annotation></semantics></math> is still locally free of rank <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math> as can be seen by taking intersections of the trivializing covers. So <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Pic</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Pic(X)</annotation></semantics></math> is closed under tensor product.</p> <p>There is an identity element, since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mi>X</mi></msub><mo>⊗</mo><mi>ℒ</mi><mo>≃</mo><mi>ℒ</mi></mrow><annotation encoding="application/x-tex">\mathcal{O}_X\otimes \mathcal{L}\simeq \mathcal{L}</annotation></semantics></math>. The tensor product is associative.</p> <p>Lastly, given any invertible sheaf <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">\mathcal{L}</annotation></semantics></math> we check that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℒ</mi> <mo>∧</mo></msup><mo>=</mo><mi>ℋℴ𝓂</mi><mo stretchy="false">(</mo><mi>ℒ</mi><mo>,</mo><msub><mi>𝒪</mi> <mi>X</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{L}^\wedge=\mathcal{Hom}(\mathcal{L}, \mathcal{O}_X)</annotation></semantics></math> is its inverse. Consider <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℒ</mi> <mo>∧</mo></msup><mo>⊗</mo><mi>ℒ</mi><mo>≃</mo><mi>ℋℴ𝓂</mi><mo stretchy="false">(</mo><mi>ℒ</mi><mo>,</mo><mi>ℒ</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>𝒪</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{L}^\wedge \otimes \mathcal{L}\simeq \mathcal{Hom}(\mathcal{L}, \mathcal{L})\simeq \mathcal{O}_X</annotation></semantics></math>.</p> </div> <h3 id="in_algebraic_geometry">In algebraic geometry</h3> <p>Suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is an integral <a class="existingWikiWord" href="/nlab/show/scheme">scheme</a> over a <a class="existingWikiWord" href="/nlab/show/field">field</a>. The correspondence between <a class="existingWikiWord" href="/nlab/show/Cartier+divisors">Cartier divisors</a> and <span class="newWikiWord">invertible sheaves<a href="/nlab/new/invertible+sheaves">?</a></span> is given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>↦</mo><msub><mi>𝒪</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D\mapsto \mathcal{O}_X(D)</annotation></semantics></math>. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> is represented by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{(U_i, f_i)\}</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}_X(D)</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{O}_X</annotation></semantics></math>-submodule of <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">\mathcal{K}</annotation></semantics></math>, the sheaf of quotients, generated by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>f</mi> <mi>i</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">f_i^{-1}</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math>. Under our assumptions, this map is an isomorphism between the Cartier class divisor group and Picard group, but for a general scheme it is only injective. Under the additional assumptions that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/separated+scheme">separated</a> and locally factorial, we get an isomorphism between the class divisor group and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Pic</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Pic(X)</annotation></semantics></math>.</p> <p>Another form the Picard group takes is from the isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Pic</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>H</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msubsup><mi>𝒪</mi> <mi>X</mi> <mo>*</mo></msubsup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Pic(X)\simeq H^1(X, \mathcal{O}_X^*)</annotation></semantics></math>. The isomorphism is most easily seen by looking at the transition functions for a trivializing cover of <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">\mathcal{L}</annotation></semantics></math>. Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\phi_i)</annotation></semantics></math> trivialize <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">\mathcal{L}</annotation></semantics></math> over the cover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(U_i)</annotation></semantics></math>. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>ϕ</mi> <mi>i</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>∘</mo><msub><mi>ϕ</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">\phi_i^{-1}\circ \phi_j</annotation></semantics></math> is an automorphism of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{O}_{U_i\cap U_j}</annotation></semantics></math>, i.e. a section of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝒪</mi> <mi>X</mi> <mo>*</mo></msubsup><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}_X^*(U_i\cap U_j)</annotation></semantics></math>. One can check this defines a <a class="existingWikiWord" href="/nlab/show/%3Fech+cohomology">?ech cocycle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mover><mi>H</mi><mo stretchy="false">ˇ</mo></mover> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>𝒰</mi><mo>,</mo><msubsup><mi>𝒪</mi> <mi>X</mi> <mo>*</mo></msubsup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\check{H}^1(\mathcal{U}, \mathcal{O}_X^*)</annotation></semantics></math> which is isomorphic to the abelian <a class="existingWikiWord" href="/nlab/show/sheaf+cohomology">sheaf cohomology</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msubsup><mi>𝒪</mi> <mi>X</mi> <mo>*</mo></msubsup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^1(X, \mathcal{O}_X^*)</annotation></semantics></math>.</p> <h2 id="examples">Examples</h2> <ul> <li>over a <a class="existingWikiWord" href="/nlab/show/Riemann+surface">Riemann surface</a>: see at <em><a href="Riemann+surface#PicardGroupOfHolomorphicLineBundles">Riemann surface – Picard group of holomorphic line bundles</a></em>.</li> </ul> <h2 id="properties">Properties</h2> <ul> <li>Let <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> be the <a class="existingWikiWord" href="/nlab/show/ring+of+integers">ring of integers</a> of an algebraic number field. Then <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> is a <a class="existingWikiWord" href="/nlab/show/unique+factorization+domain">unique factorization domain</a> if and only if the Picard group 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> is <a class="existingWikiWord" href="/nlab/show/trivial+group">trivial</a>.</li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a>, <strong>Picard group</strong>, <a class="existingWikiWord" href="/nlab/show/Picard+stack">Picard stack</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Brauer+group">Brauer group</a>, <a class="existingWikiWord" href="/nlab/show/Azumaya+algebra">Azumaya algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Picard+2-group">Picard 2-group</a>, <a class="existingWikiWord" href="/nlab/show/Picard+%E2%88%9E-group">Picard ∞-group</a>, <a class="existingWikiWord" href="/nlab/show/Picard+%E2%88%9E-stack">Picard ∞-stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Picard+scheme">Picard scheme</a></p> </li> <li> <p><span class="newWikiWord">Tate?Shafarevich group<a href="/nlab/new/Tate%3FShafarevich+group">?</a></span></p> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/moduli+spaces">moduli spaces</a> of <a class="existingWikiWord" href="/nlab/show/line+n-bundles+with+connection">line n-bundles with connection</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/dimension">dimensional</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></strong></p> <table><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></th><th><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+object">Calabi-Yau n-fold</a></th><th><a class="existingWikiWord" href="/nlab/show/line+n-bundle">line n-bundle</a></th><th>moduli of <a class="existingWikiWord" href="/nlab/show/line+n-bundles">line n-bundles</a></th><th>moduli of <a class="existingWikiWord" href="/nlab/show/flat+infinity-connection">flat</a>/degree-0 n-bundles</th><th><a class="existingWikiWord" href="/nlab/show/Artin-Mazur+formal+group">Artin-Mazur formal group</a> of <a class="existingWikiWord" href="/nlab/show/deformation+theory">deformation moduli</a> of <a class="existingWikiWord" href="/nlab/show/line+n-bundles">line n-bundles</a></th><th><a class="existingWikiWord" href="/nlab/show/complex+oriented+cohomology+theory">complex oriented cohomology theory</a></th><th><a class="existingWikiWord" href="/nlab/show/modular+functor">modular functor</a>/<a class="existingWikiWord" href="/nlab/show/self-dual+higher+gauge+theory">self-dual higher gauge theory</a> of <a class="existingWikiWord" href="/nlab/show/higher+dimensional+Chern-Simons+theory">higher dimensional Chern-Simons theory</a></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n = 0</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/unit">unit</a> in <a class="existingWikiWord" href="/nlab/show/structure+sheaf">structure sheaf</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/multiplicative+group">multiplicative group</a>/<a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/formal+multiplicative+group">formal multiplicative group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/complex+K-theory">complex K-theory</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><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></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/elliptic+curve">elliptic curve</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/line+bundle">line bundle</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Picard+group">Picard group</a>/<a class="existingWikiWord" href="/nlab/show/Picard+scheme">Picard scheme</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Jacobian">Jacobian</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/formal+Picard+group">formal Picard group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/3d+Chern-Simons+theory">3d Chern-Simons theory</a>/<a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n = 2</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/K3+surface">K3 surface</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/line+2-bundle">line 2-bundle</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Brauer+group">Brauer group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/intermediate+Jacobian">intermediate Jacobian</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/formal+Brauer+group">formal Brauer group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/K3+cohomology">K3 cohomology</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">n = 3</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+3-fold">Calabi-Yau 3-fold</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/line+3-bundle">line 3-bundle</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/intermediate+Jacobian">intermediate Jacobian</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+cohomology">CY3 cohomology</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/7d+Chern-Simons+theory">7d Chern-Simons theory</a>/<a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a></td></tr> <tr><td style="text-align: left;"><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></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/intermediate+Jacobian">intermediate Jacobian</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <h3 id="general">General</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, <em>Hoàng Xuân Sính’s thesis: categorifying group theory</em> (<a href="https://math.ucr.edu/home/baez/sinh.pdf">pdf</a>)</li> </ul> <h3 id="in_algebraic_geometry_2">In algebraic geometry</h3> <ul> <li>Robin Hartshorne, Algebraic Geometry</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on August 7, 2023 at 00:47:11. See the <a href="/nlab/history/Picard+group" 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/Picard+group" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/3005/#Item_10">Discuss</a><span class="backintime"><a href="/nlab/revision/Picard+group/15" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/Picard+group" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/Picard+group" accesskey="S" class="navlink" id="history" rel="nofollow">History (15 revisions)</a> <a href="/nlab/show/Picard+group/cite" style="color: black">Cite</a> <a href="/nlab/print/Picard+group" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/Picard+group" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>