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> extension of scalars 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> extension of scalars </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/4266/#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="algebra">Algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a></p> <h2 id="algebraic_theories">Algebraic theories</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+theory">algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/2-algebraic+theory">2-algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-algebraic+theory">(∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad">monad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-monad">(∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operad">operad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-operad">(∞,1)-operad</a></p> </li> </ul> <h2 id="algebras_and_modules">Algebras and modules</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+a+monad">algebra over a monad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-monad">∞-algebra over an (∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+algebraic+theory">algebra over an algebraic theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-algebraic+theory">∞-algebra over an (∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over an operad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-operad">∞-algebra over an (∞,1)-operad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a></p> </li> </ul> <h2 id="higher_algebras">Higher algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+in+an+%28%E2%88%9E%2C1%29-category">monoid in an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+an+%28%E2%88%9E%2C1%29-category">commutative monoid in an (∞,1)-category</a></p> </li> </ul> </li> <li> <p>symmetric monoidal (∞,1)-category of spectra</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+smash+product+of+spectra">symmetric monoidal smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</a>, <a class="existingWikiWord" href="/nlab/show/module+spectrum">module spectrum</a>, <a class="existingWikiWord" href="/nlab/show/algebra+spectrum">algebra spectrum</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebra">A-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+space">A-∞ space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/C-%E2%88%9E+algebra">C-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+algebra">E-∞ algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module+bundle">(∞,1)-module bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative cohomology theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/deformation+theory">deformation theory</a></li> </ul> </li> </ul> <h2 id="model_category_presentations">Model category presentations</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+T-algebras">model structure on simplicial T-algebras</a> / <a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+operads">model structure on operads</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">model structure on algebras over an operad</a></p> </li> </ul> <h2 id="geometry_on_formal_duals_of_algebras">Geometry on formal duals of algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+conjecture">Deligne conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/higher+algebra+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#general_abstract'>General abstract</a></li> <li><a href='#in_components'>In components</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#geometric_interpretation'>Geometric interpretation</a></li> <li><a href='#frobenius_extensions'>Frobenius extensions</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The <em>extension of scalars</em> of a <a class="existingWikiWord" href="/nlab/show/module">module</a> along a <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of <a class="existingWikiWord" href="/nlab/show/rings">rings</a> is the <a class="existingWikiWord" href="/nlab/show/Isbell+duality">algebraic dual</a> of what geometrically is the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> of <a class="existingWikiWord" href="/nlab/show/bundles">bundles</a> along a map of their base spaces (with respect to the discussion at <em><a href="http://ncatlab.org/nlab/show/module#RelationToVectorBundlesInIntroduction">modules - as generalized vector bundles</a></em>).</p> <p>Explicitly, extension of scalars along a <a class="existingWikiWord" href="/nlab/show/ring">ring</a> <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>R</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">f : R \to S</annotation></semantics></math> is the operation on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/modules">modules</a> given by forming the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+modules">tensor product of modules</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> regarded as an <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>-module via <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>.</p> <p>There are similar functors for <a class="existingWikiWord" href="/nlab/show/bimodules">bimodules</a> and in some other categories.</p> <h2 id="definition">Definition</h2> <p>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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/commutative+rings">commutative rings</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>R</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">f \colon R\to S</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of <a class="existingWikiWord" href="/nlab/show/rings">rings</a>.</p> <p>We discuss <em>extension of scalars</em> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> first <a class="existingWikiWord" href="/nlab/show/category+theory">general abstractly</a> and then explicitly in components.</p> <h3 id="general_abstract">General abstract</h3> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Mod">Mod</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Mod">Mod</a> for the <a class="existingWikiWord" href="/nlab/show/categories+of+modules">categories of modules</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>, respectively.</p> <div class="num_defn" id="RestrictionOfScalars"> <h6 id="definition_2">Definition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/ring">ring</a> <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>R</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">f : R \to S</annotation></semantics></math> the <strong><a class="existingWikiWord" href="/nlab/show/restriction+of+scalars">restriction of scalars</a></strong> functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo>:</mo><mi>S</mi><mi>Mod</mi><mo>→</mo><mi>R</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex"> f^* : S Mod \to R Mod </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/functor">functor</a> that takes an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/module">module</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math> to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-module <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mi>N</mi></mrow><annotation encoding="application/x-tex">f^*N</annotation></semantics></math> whose underlying <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> is that of <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> and whose <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/action">action</a> is given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>⋅</mo><mi>n</mi><mo>≔</mo><mi>f</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>n</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>for</mi><mspace width="thickmathspace"></mspace><mi>r</mi><mo>∈</mo><mi>R</mi><mo>,</mo><mi>n</mi><mo>∈</mo><mi>N</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> r \cdot n \coloneqq f(r)\cdot n \;\;\;\; for \; r \in R, n \in N \,. </annotation></semantics></math></div></div> <div class="num_prop" id="AdjointPair"> <h6 id="proposition">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/restriction+of+scalars">restriction of scalars</a> functor, def. <a class="maruku-ref" href="#RestrictionOfScalars"></a>, is the <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> in a pair of <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>f</mi> <mo>!</mo></msub><mo>⊣</mo><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>S</mi><mi>Mod</mi><mover><munder><mo>⟶</mo><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow></munder><mover><mo>⟵</mo><mrow><msub><mi>f</mi> <mo>!</mo></msub></mrow></mover></mover><mi>R</mi><mi>Mod</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> ( f_! \dashv f^* ) \;\colon\; S Mod \stackrel{\overset{f_!}{\longleftarrow}} {\underset{f^*}{\longrightarrow}} R Mod \,. </annotation></semantics></math></div></div> <div class="num_defn" id="ExtensionByAdjoint"> <h6 id="definition_3">Definition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>!</mo></msub><mo lspace="verythinmathspace">:</mo><mi>R</mi><mi>Mod</mi><mo>→</mo><mi>S</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">f_! \colon R Mod \to S Mod</annotation></semantics></math> in prop. <a class="maruku-ref" href="#AdjointPair"></a> is called <strong>extension of scalars</strong> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>A further <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">f_*</annotation></semantics></math> would be called <em><a class="existingWikiWord" href="/nlab/show/coextension+of+scalars">coextension of scalars</a></em> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>.</p> </div> <p>See also <a class="existingWikiWord" href="/nlab/show/induced+representation">induced representation</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">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/restricted+representation">restricted representation</a>.</p> <h3 id="in_components">In components</h3> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/ring">ring</a> <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>R</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">f : R \to S</annotation></semantics></math>, the <em>extension of scalars</em> <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>!</mo></msub></mrow><annotation encoding="application/x-tex">f_!</annotation></semantics></math> of def. <a class="maruku-ref" href="#ExtensionByAdjoint"></a> is the functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>!</mo></msub><mo>≔</mo><mi>S</mi><msub><mo>⊗</mo> <mi>R</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>:</mo><mspace width="thinmathspace"></mspace><mi>R</mi><mi>Mod</mi><mo>→</mo><mi>S</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex"> f_! \coloneqq S \otimes_R (-) \,:\, R Mod \to S Mod </annotation></semantics></math></div> <p>given by <a class="existingWikiWord" href="/nlab/show/tensor+product+of+modules">tensor product of modules</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> regarded as an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-<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/bimodule">bimodule</a>: the left <a class="existingWikiWord" href="/nlab/show/action">action</a> being the canonical action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> on itself, the right being the <a class="existingWikiWord" href="/nlab/show/restriction+of+scalars">restriction of scalars</a>-action along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>.</p> <p>Explicitly, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>∈</mo><mi>R</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">N \in R Mod</annotation></semantics></math></p> <ul> <li> <p>the elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>!</mo></msub><mi>N</mi></mrow><annotation encoding="application/x-tex">f_! N</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/equivalence+classes">equivalence classes</a> of pairs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>S</mi><mo>×</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">(s,n) \in S \times N</annotation></semantics></math> under the <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>s</mi><mo>⋅</mo><mi>f</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mi>r</mi><mo>⋅</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (s \cdot f(r), n) = (s, r\cdot n) </annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">s \in S</annotation></semantics></math>;</p> </li> <li> <p>the left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/action">action</a> is given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>′</mo><mo>⋅</mo><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>s</mi><mo>′</mo><mo>⋅</mo><mi>s</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">s' \cdot(s,n) = (s' \cdot s,n)</annotation></semantics></math>.</p> </li> </ul> </div> <h2 id="properties">Properties</h2> <h3 id="geometric_interpretation">Geometric interpretation</h3> <p>Under <a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a> extension of scalars turns into a statement about <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a>.</p> <p>By definition the category</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ring</mi> <mi>op</mi></msup><munderover><mo>→</mo><mi>Spec</mi><mrow><mo lspace="verythinmathspace">:</mo><mo>≃</mo></mrow></munderover><mi>Aff</mi></mrow><annotation encoding="application/x-tex"> Ring^{op} \underoverset{Spec}{\colon \simeq}{\to} Aff </annotation></semantics></math></div> <p>of (absolute) <a class="existingWikiWord" href="/nlab/show/affine+schemes">affine schemes</a> is the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> of <a class="existingWikiWord" href="/nlab/show/Ring">Ring</a>.</p> <p>Hence for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>R</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">f : R \to S</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/ring">ring</a> <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a>, we have equivalently a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Spec(f) : Spec(S) \to Spec(R) </annotation></semantics></math></div> <p>of affine schemes.</p> <p>An <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/module">module</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math> corresponds to the collection of <a class="existingWikiWord" href="/nlab/show/sections">sections</a> of a “generalized <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a>” over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(R)</annotation></semantics></math>: something that has a <a class="existingWikiWord" href="/nlab/show/quasicoherent+sheaf">quasicoherent sheaf</a> of sections.</p> <p>The <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> of this “bundle” along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(f)</annotation></semantics></math> has sections forming the module <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>!</mo></msub><mi>N</mi></mrow><annotation encoding="application/x-tex">f_! N</annotation></semantics></math>.</p> <p>Generally, for any <a class="existingWikiWord" href="/nlab/show/fibered+category">fibered category</a> like <a class="existingWikiWord" href="/nlab/show/Mod">Mod</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo><mi>Aff</mi></mrow><annotation encoding="application/x-tex">\to Aff</annotation></semantics></math> we may regard the <a class="existingWikiWord" href="/nlab/show/inverse+image+functor">inverse image functor</a> as the extension of scalars.</p> <p>For that reason if there is some other fibered category <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">\mathcal{F}</annotation></semantics></math> over the opposite of some algebraic category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> whose objects are considered “objects of scalars” one is inclined to call the inverse image functor, the extension of scalars.</p> <h3 id="frobenius_extensions">Frobenius extensions</h3> <p>Extension of scalars coincides with <em><a class="existingWikiWord" href="/nlab/show/coextension+of+scalars">coextension of scalars</a></em> (to make an <a class="existingWikiWord" href="/nlab/show/ambidextrous+adjunction">ambidextrous adjunction</a> with <a class="existingWikiWord" href="/nlab/show/restriction+of+scalars">restriction of scalars</a>) in the case of <a class="existingWikiWord" href="/nlab/show/Frobenius+extensions">Frobenius extensions</a>, see there for more.</p> <h2 id="examples">Examples</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/complexification">complexification</a> is extension of scalars along the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mo>↪</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R} \hookrightarrow \mathbb{C}</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> into the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a>.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localization+of+a+module">localization of a module</a></p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><strong>extension of scalars</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/restriction+of+scalars">restriction of scalars</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">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/coextension+of+scalars">coextension of scalars</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extension+%28double+category+theory%29">extension (double category theory)</a></p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on August 14, 2024 at 12:03:51. See the <a href="/nlab/history/extension+of+scalars" 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/extension+of+scalars" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/4266/#Item_1">Discuss</a><span class="backintime"><a href="/nlab/revision/extension+of+scalars/10" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/extension+of+scalars" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/extension+of+scalars" accesskey="S" class="navlink" id="history" rel="nofollow">History (10 revisions)</a> <a href="/nlab/show/extension+of+scalars/cite" style="color: black">Cite</a> <a href="/nlab/print/extension+of+scalars" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/extension+of+scalars" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>