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href='/nlab/show/diff/2-Lawvere+theory'>2-algebraic theory</a> / <a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-algebraic+theory'>(∞,1)-algebraic theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monad'>monad</a> / <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-monad'>(∞,1)-monad</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/operad'>operad</a> / <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%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/diff/algebra+over+a+monad'>algebra over a monad</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/infinity-algebra+over+an+%28infinity%2C1%29-monad'>∞-algebra over an (∞,1)-monad</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/algebra+over+a+Lawvere+theory'>algebra over an algebraic theory</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/infinity-algebra+over+an+%28infinity%2C1%29-algebraic+theory'>∞-algebra over an (∞,1)-algebraic theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/algebra+over+an+operad'>algebra over an operad</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/infinity-algebra+over+an+%28infinity%2C1%29-operad'>∞-algebra over an (∞,1)-operad</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/action'>action</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-action'>∞-action</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/representation'>representation</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-representation'>∞-representation</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/module'>module</a>, <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-module'>∞-module</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/associated+bundle'>associated bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/associated+infinity-bundle'>associated ∞-bundle</a></p> </li> </ul> <h2 id='higher_algebras'>Higher algebras</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monoidal+%28infinity%2C1%29-category'>monoidal (∞,1)-category</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+%28infinity%2C1%29-category'>symmetric monoidal (∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monoid+in+a+monoidal+%28infinity%2C1%29-category'>monoid in an (∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/commutative+monoid+in+a+symmetric+monoidal+%28infinity%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/diff/smash+product+of+spectra'>smash product of spectra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/symmetric+smash+product+of+spectra'>symmetric monoidal smash product of spectra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/ring+spectrum'>ring spectrum</a>, <a class='existingWikiWord' href='/nlab/show/diff/module+spectrum'>module spectrum</a>, <a class='existingWikiWord' href='/nlab/show/diff/algebra+spectrum'>algebra spectrum</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/A-infinity-algebra'>A-∞ algebra</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/A-infinity-ring'>A-∞ ring</a>, <a class='existingWikiWord' href='/nlab/show/diff/A-infinity-space'>A-∞ space</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/C_%E2%88%9E-algebra'>C-∞ algebra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/E-infinity-ring'>E-∞ ring</a>, <a class='existingWikiWord' href='/nlab/show/diff/E-infinity+algebra'>E-∞ algebra</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-module'>∞-module</a>, <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-module+bundle'>(∞,1)-module bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/multiplicative+cohomology+theory'>multiplicative cohomology theory</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/L-infinity-algebra'>L-∞ algebra</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/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/diff/model+structure+on+simplicial+algebras'>model structure on simplicial T-algebras</a> / <a class='existingWikiWord' href='/nlab/show/diff/homotopy+T-algebra'>homotopy T-algebra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+operads'>model structure on operads</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/Isbell+duality'>Isbell duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/derived+geometry'>derived geometry</a></p> </li> </ul> <h2 id='theorems'>Theorems</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Deligne+conjecture'>Deligne conjecture</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/delooping+hypothesis'>delooping hypothesis</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#definition'>Definition</a><ul><li><a href='#general_abstract'>General abstract</a></li><li><a href='#in_components'>In components</a></li></ul></li><li><a href='#properties'>Properties</a><ul><li><a href='#geometric_interpretation'>Geometric interpretation</a></li></ul></li><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/diff/module'>module</a> along a <a class='existingWikiWord' href='/nlab/show/diff/homomorphism'>homomorphism</a> of <a class='existingWikiWord' href='/nlab/show/diff/ring'>rings</a> is the <a class='existingWikiWord' href='/nlab/show/diff/Isbell+duality'>algebraic dual</a> of what geometrically is the <a class='existingWikiWord' href='/nlab/show/diff/pullback'>pullback</a> of <a class='existingWikiWord' href='/nlab/show/diff/bundle'>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/diff/ring'>ring</a> <a class='existingWikiWord' href='/nlab/show/diff/homomorphism'>homomorphism</a> <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>S</mi><mo>→</mo><mi>R</mi></mrow><annotation encoding='application/x-tex'>f : S \to R</annotation></semantics></math> is the operation on <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/module'>modules</a> given by forming the <a class='existingWikiWord' href='/nlab/show/diff/tensor+product+of+modules'>tensor product of modules</a> with <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> regarded as an <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>-module via <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_5' xmlns='http://www.w3.org/1998/Math/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/diff/bimodule'>bimodules</a> and in some other categories.</p> <h2 id='definition'>Definition</h2> <p>Let <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> be <a class='existingWikiWord' href='/nlab/show/diff/ring'>commutative rings</a> and let <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_8' xmlns='http://www.w3.org/1998/Math/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/diff/homomorphism'>homomorphism</a> of <a class='existingWikiWord' href='/nlab/show/diff/ring'>rings</a>.</p> <p>We discuss <em>extension of scalars</em> along <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> first <a class='existingWikiWord' href='/nlab/show/diff/category+theory'>general abstractly</a> and then explicitly in components.</p> <h3 id='general_abstract'>General abstract</h3> <p>Write <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/Mod'>Mod</a> and <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/Mod'>Mod</a> for the <a class='existingWikiWord' href='/nlab/show/diff/Mod'>categories of modules</a> over <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_13' xmlns='http://www.w3.org/1998/Math/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/diff/ring'>ring</a> <a class='existingWikiWord' href='/nlab/show/diff/homomorphism'>homomorphism</a> <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_14' xmlns='http://www.w3.org/1998/Math/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/diff/restriction+of+scalars'>restriction of scalars</a></strong> functor</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_15' xmlns='http://www.w3.org/1998/Math/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/diff/functor'>functor</a> that takes an <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/module'>module</a> <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>N</mi></mrow><annotation encoding='application/x-tex'>N</annotation></semantics></math> to the <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>-module <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_19' xmlns='http://www.w3.org/1998/Math/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/diff/abelian+group'>abelian group</a> is that of <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>N</mi></mrow><annotation encoding='application/x-tex'>N</annotation></semantics></math> and whose <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/action'>action</a> is given by</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_22' xmlns='http://www.w3.org/1998/Math/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 width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mi>for</mi><mi>r</mi><mo>∈</mo><mi>R</mi><mo>,</mo><mi>n</mi><mo>∈</mo><mi>N</mi><mspace width='thinmathspace' /><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/diff/restriction+of+scalars'>restriction of scalars</a> functor, def. <a class='maruku-ref' href='#RestrictionOfScalars'>1</a>, is the <a class='existingWikiWord' href='/nlab/show/diff/right+adjoint'>right adjoint</a> in a pair of <a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor'>adjoint functors</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_23' xmlns='http://www.w3.org/1998/Math/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><mo>:</mo><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' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> ( f_! \dashv f^* ) : S Mod \stackrel{\overset{f_!}{\leftarrow}}{\underset{f^*}{\to}} 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/diff/left+adjoint'>left adjoint</a> <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_24' xmlns='http://www.w3.org/1998/Math/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'>1</a> is called <strong>extension of scalars</strong> along <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_25' xmlns='http://www.w3.org/1998/Math/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/diff/right+adjoint'>right adjoint</a> <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_26' xmlns='http://www.w3.org/1998/Math/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/diff/coextension+of+scalars'>coextension of scalars</a></em> along <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math>.</p> </div> <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/diff/ring'>ring</a> <a class='existingWikiWord' href='/nlab/show/diff/homomorphism'>homomorphism</a> <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_28' xmlns='http://www.w3.org/1998/Math/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/diff/functor'>functor</a> <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_29' xmlns='http://www.w3.org/1998/Math/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'>2</a> is the functor</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_30' xmlns='http://www.w3.org/1998/Math/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' /><mo>:</mo><mspace width='thinmathspace' /><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/diff/tensor+product+of+modules'>tensor product of modules</a> with <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> regarded as an <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>-<math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/bimodule'>bimodule</a>: the left <a class='existingWikiWord' href='/nlab/show/diff/action'>action</a> being the canonical action of <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_34' xmlns='http://www.w3.org/1998/Math/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/diff/restriction+of+scalars'>restriction of scalars</a>-action along <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math>.</p> <p>Explicitly, for <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>N</mi><mo>∈</mo><mi>S</mi><mi>Mod</mi></mrow><annotation encoding='application/x-tex'>N \in S Mod</annotation></semantics></math></p> <ul> <li> <p>the elements of <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_37' xmlns='http://www.w3.org/1998/Math/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/diff/equivalence+class'>equivalence classes</a> of pairs <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_38' xmlns='http://www.w3.org/1998/Math/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/diff/equivalence+relation'>equivalence relation</a> <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_39' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_40' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/action'>action</a> is given by <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_42' xmlns='http://www.w3.org/1998/Math/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/diff/Isbell+duality'>Isbell duality</a> extension of scalars turns into a statement about <a class='existingWikiWord' href='/nlab/show/diff/geometry'>geometry</a>.</p> <p>By definition the category</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_43' xmlns='http://www.w3.org/1998/Math/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/diff/affine+scheme'>affine schemes</a> is the <a class='existingWikiWord' href='/nlab/show/diff/opposite+category'>opposite category</a> of <a class='existingWikiWord' href='/nlab/show/diff/Ring'>Ring</a>.</p> <p>Hence for <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_44' xmlns='http://www.w3.org/1998/Math/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/diff/ring'>ring</a> <a class='existingWikiWord' href='/nlab/show/diff/homomorphism'>homomorphism</a>, we have equivalently a morphism</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_45' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/module'>module</a> <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_47' xmlns='http://www.w3.org/1998/Math/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/diff/section'>sections</a> of a “generalized <a class='existingWikiWord' href='/nlab/show/diff/vector+bundle'>vector bundle</a>” over <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_48' xmlns='http://www.w3.org/1998/Math/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/diff/quasicoherent+sheaf'>quasicoherent sheaf</a> of sections.</p> <p>The <a class='existingWikiWord' href='/nlab/show/diff/pullback'>pullback</a> of this “bundle” along <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_49' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_50' xmlns='http://www.w3.org/1998/Math/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/diff/Grothendieck+fibration'>fibered category</a> like <a class='existingWikiWord' href='/nlab/show/diff/Mod'>Mod</a><math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_51' xmlns='http://www.w3.org/1998/Math/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/diff/inverse+image'>inverse image functor</a> as the extension of scalars.</p> <p>For that reason if there is some other fibered category <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_52' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_53' xmlns='http://www.w3.org/1998/Math/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> <h2 id='examples'>Examples</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/complexification'>complexification</a> is extension of scalars along the inclusion <math class='maruku-mathml' display='inline' id='mathml_19c9da06e2486513a20d6af5cce816aeb0f13f28_54' xmlns='http://www.w3.org/1998/Math/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/diff/real+number'>real numbers</a> into the <a class='existingWikiWord' href='/nlab/show/diff/complex+number'>complex numbers</a>.</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/localization+of+a+module'>localization of a module</a></p> </li> </ul> <h2 id='related_concepts'>Related concepts</h2> <ul> <li><ins class='diffins'><strong>extension of scalars</strong></ins><ins class='diffins'> </ins><ins class='diffins'><math class='maruku-mathml' display='inline' id='mathml_a1197e5530a260e68e36a1e1387eb5240117dcda_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊣</mo></mrow><annotation encoding='application/x-tex'>\dashv</annotation></semantics></math></ins><ins class='diffins'> </ins><ins class='diffins'><a class='existingWikiWord' href='/nlab/show/diff/restriction+of+scalars'>restriction of scalars</a></ins><ins class='diffins'> </ins><ins class='diffins'><math class='maruku-mathml' display='inline' id='mathml_a1197e5530a260e68e36a1e1387eb5240117dcda_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊣</mo></mrow><annotation encoding='application/x-tex'>\dashv</annotation></semantics></math></ins><ins class='diffins'> </ins><a class='existingWikiWord' href='/nlab/show/diff/coextension+of+scalars'>coextension of scalars</a></li> </ul> </div>  <div class="revisedby"> <p> Revision on May 30, 2013 at 18:03:35 by <a href="/nlab/author/Urs+Schreiber" style="color: #005c19">Urs Schreiber</a> 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="https://nforum.ncatlab.org/discussion/4266/#Item_1">Discuss</a><span class="backintime"><a href="/nlab/revision/diff/extension+of+scalars/5" accesskey="F" class="navlinkbackintime" id="to_next_revision" rel="nofollow">Next revision</a> (7 more)</span><span class="backintime"><a href="/nlab/revision/diff/extension+of+scalars/3" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a> (3 more)</span><a href="/nlab/show/diff/extension+of+scalars" class="navlink" id="to_current_revision">Current version of page</a><a href="/nlab/revision/extension+of+scalars/4" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Hide changes</a><a href="/nlab/history/extension+of+scalars" accesskey="S" class="navlink" id="history" rel="nofollow">History (10 revisions)</a><a href="/nlab/rollback/extension+of+scalars?rev=4" class="navlink" id="rollback" rel="nofollow">Rollback</a> <a href="/nlab/revision/extension+of+scalars/4/cite" style="color: black">Cite</a> <a href="/nlab/source/extension+of+scalars/4" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>