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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='#properties'>Properties</a></li> <ul> <li><a href='#RelationToCatsOfModules'>Relation to categories of modules</a></li> <li><a href='#RelationToEtaleCohomology'>Relation to étale cohomology</a></li> <li><a href='#RelationToDerivedEtaleCohomology'>Relation to derived étale cohomology</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>For <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 <a class="existingWikiWord" href="/nlab/show/ring">ring</a>, the <em>Brauer group</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Br</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Br(R)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/group">group</a> of <a class="existingWikiWord" href="/nlab/show/Morita+equivalence">Morita equivalence</a> classes of <a class="existingWikiWord" href="/nlab/show/Azumaya+algebras">Azumaya algebras</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>.</p> <h2 id="properties">Properties</h2> <h3 id="RelationToCatsOfModules">Relation to categories of modules</h3> <div class="num_defn"> <h6 id="definition">Definition</h6> <p>For <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 commutative <a class="existingWikiWord" href="/nlab/show/ring">ring</a>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Alg</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">Alg_R</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><msub><mi>Vect</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">2Vect_R</annotation></semantics></math> (see at <a class="existingWikiWord" href="/nlab/show/2-vector+space">2-vector space</a>/<a class="existingWikiWord" href="/nlab/show/2-module">2-module</a>) be the <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> whose</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/objects">objects</a> are <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/associative+algebra">algebras</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> are <a class="existingWikiWord" href="/nlab/show/bimodules">bimodules</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>-algebras;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-morphisms">2-morphisms</a> are bimodule homomorphisms.</p> </li> </ul> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>This may be understood as the 2-category of (generalized) <a class="existingWikiWord" href="/nlab/show/2-vector+bundles">2-vector bundles</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mi>R</mi></mrow><annotation encoding="application/x-tex">Spec R</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/Isbell+duality">formally dual</a> <a class="existingWikiWord" href="/nlab/show/space">space</a> whose <a class="existingWikiWord" href="/nlab/show/function+algebra">function algebra</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>. This is a <a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal 2-category</a>.</p> </div> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Br</mi></mstyle><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>Core</mi><mo stretchy="false">(</mo><msub><mi>Alg</mi> <mi>R</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{Br}(R) \coloneqq Core(Alg_R) </annotation></semantics></math></div> <p>be its <a class="existingWikiWord" href="/nlab/show/Picard+3-group">Picard 3-group</a>, hence the maximal <a class="existingWikiWord" href="/nlab/show/infinity-group">3-group</a> inside (which is hence a <a class="existingWikiWord" href="/nlab/show/braided+3-group">braided 3-group</a>), the <a class="existingWikiWord" href="/nlab/show/core">core</a> on the invertible objects, hence the <a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a> whose</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/objects">objects</a> are algebras which are invertible up to <a class="existingWikiWord" href="/nlab/show/Morita+equivalence">Morita equivalence</a> under tensor product;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> are <a class="existingWikiWord" href="/nlab/show/Morita+equivalences">Morita equivalences</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-morphisms">2-morphisms</a> are invertible bimodule homomorphisms.</p> </li> </ul> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>This may be understood as the 2-groupoid of (generalized) <a class="existingWikiWord" href="/nlab/show/line+2-bundles">line 2-bundles</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mi>R</mi></mrow><annotation encoding="application/x-tex">Spec R</annotation></semantics></math> (for instance <a class="existingWikiWord" href="/nlab/show/holomorphic+line+2-bundles">holomorphic line 2-bundles</a> in the case of <a class="existingWikiWord" href="/nlab/show/higher+complex+analytic+geometry">higher complex analytic geometry</a>), inside that of all <a class="existingWikiWord" href="/nlab/show/2-vector+bundles">2-vector bundles</a>.</p> </div> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Br</mi></mstyle><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{Br}(R)</annotation></semantics></math> are the following:</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>Br</mi></mstyle><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_0(\mathbf{Br}(R))</annotation></semantics></math> is the Brauer 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>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>Br</mi></mstyle><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(\mathbf{Br}(R))</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Picard+group">Picard group</a> 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>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>Br</mi></mstyle><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_2(\mathbf{Br}(R))</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a> 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>.</p> </li> </ul> </div> <p>See for instance (<a href="#Street">Street</a>).</p> <div class="num_example"> <h6 id="example">Example</h6> <p>Analogous statements hold for (non-commutative) <a class="existingWikiWord" href="/nlab/show/superalgebras">superalgebras</a>, hence for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/graded+algebras">graded algebras</a>. See at <em><a href="super+algebra#Picard2Groupoid">superalgebra – Picard 3-group, Brauer group</a></em>.</p> </div> <h3 id="RelationToEtaleCohomology">Relation to étale cohomology</h3> <p>The Brauer group of a <a class="existingWikiWord" href="/nlab/show/ring">ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/torsion">torsion</a> subgroup of the second <a class="existingWikiWord" href="/nlab/show/etale+cohomology">etale cohomology</a> group of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mi>R</mi></mrow><annotation encoding="application/x-tex">Spec R</annotation></semantics></math> with values in the <a class="existingWikiWord" href="/nlab/show/multiplicative+group">multiplicative group</a> <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></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Br</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>↪</mo><msubsup><mi>H</mi> <mi>et</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>𝔾</mi> <mi>m</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Br(X) \hookrightarrow H^2_{et}(X, \mathbb{G}_m) \,. </annotation></semantics></math></div> <p>This was first stated in (<a href="#Grothendieck68">Grothendieck 68</a>) (see also <a href="#Grothendieck64">Grothendieck 64, prop. 1.4</a> and see at <em><a href="algebraic+line+n-bundle#Properties">algebraic line n-bundle – Properties</a></em>). Review discussion is in (<a href="#Milne">Milne, chapter IV</a>). A detailed discussion in the context of <a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a> is in (<a href="#Giraud">Giraud</a>).</p> <p>A theorem stating conditions under which the Brauer group is precisely the <a class="existingWikiWord" href="/nlab/show/torsion">torsion</a> subgroup of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>et</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>𝔾</mi> <mi>m</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^2_{et}(X, \mathbb{G}_m)</annotation></semantics></math> is due to (<a href="#Gabber">Gabber</a>), see also the review in (<a href="#deJong">de Jong</a>). For more details and more literature on this see (<a href="#Bertuccioni">Bertuccioni</a>).</p> <p>This fits into the following pattern</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>et</mi> <mn>0</mn></msubsup><mo stretchy="false">(</mo><mi>R</mi><mo>,</mo><msub><mi>𝔾</mi> <mi>m</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msup><mi>R</mi> <mo>×</mo></msup></mrow><annotation encoding="application/x-tex">H^0_{et}(R, \mathbb{G}_m) = R^\times</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a>)</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>et</mi> <mn>1</mn></msubsup><mo stretchy="false">(</mo><mi>R</mi><mo>,</mo><msub><mi>𝔾</mi> <mi>m</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>Pic</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^1_{et}(R, \mathbb{G}_m) = Pic(R)</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/Picard+group">Picard group</a>: iso classes of invertible <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>-modules)</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>et</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>R</mi><mo>,</mo><msub><mi>𝔾</mi> <mi>m</mi></msub><msub><mo stretchy="false">)</mo> <mi>tor</mi></msub><mo>=</mo><mi>Br</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^2_{et}(R, \mathbb{G}_m)_{tor} = Br(R)</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/Brauer+group">Brauer group</a>: <a class="existingWikiWord" href="/nlab/show/Morita+equivalence">Morita equivalence</a> classes of <a class="existingWikiWord" href="/nlab/show/Azumaya+algebras">Azumaya algebras</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>) (the torsion equivalence classes of the <a class="existingWikiWord" href="/nlab/show/Brauer+stack">Brauer stack</a>)</p> </li> </ul> <p>It is therefore natural to regard all of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>et</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>R</mi><mo>,</mo><msub><mi>𝔾</mi> <mi>m</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^2_{et}(R, \mathbb{G}_m)</annotation></semantics></math> as the “actual” Brauer group. This has been called the “<a class="existingWikiWord" href="/nlab/show/bigger+Brauer+group">bigger Brauer group</a>” (<a href="#Taylor82">Taylor 82</a>, <a href="#CaenepeelGrandjean98">Caenepeel-Grandjean 98</a>, <a href="#HeinlothSchoeer08">Heinloth-Schöer 08</a>). The bigger Brauer group has actually traditionally been implicit already in the term “<a class="existingWikiWord" href="/nlab/show/formal+Brauer+group">formal Brauer group</a>”, which is really the <a class="existingWikiWord" href="/nlab/show/formal+geometry">formal geometry</a>-version of the bigger Brauer group.</p> <h3 id="RelationToDerivedEtaleCohomology">Relation to derived étale cohomology</h3> <p>More generally, this works for <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 (connective) <a class="existingWikiWord" href="/nlab/show/E-infinity+ring">E-infinity ring</a> (the following is due to <a href="#AntieauGepner12">Antieau-Gepner 12</a>, see <a href="#Haugseng14">Haugseng 14</a> for more).</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL_1(R)</annotation></semantics></math> be its <a class="existingWikiWord" href="/nlab/show/infinity-group+of+units">infinity-group of units</a>. If <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/connective+spectrum">connective</a>, then the first <a class="existingWikiWord" href="/nlab/show/Postnikov+tower">Postnikov stage</a> of the <a class="existingWikiWord" href="/nlab/show/Picard+group">Picard</a> <a class="existingWikiWord" href="/nlab/show/infinity-groupoid">infinity-groupoid</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Pic</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>Mod</mi><mo stretchy="false">(</mo><mi>R</mi><msup><mo stretchy="false">)</mo> <mo>×</mo></msup></mrow><annotation encoding="application/x-tex"> Pic(R) \coloneqq Mod(R)^\times </annotation></semantics></math></div> <p>is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>et</mi></msub><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Pic</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>ℤ</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}_{et} GL_1(-) &\to& Pic(-) \\ && \downarrow \\ && \mathbb{Z} } \,, </annotation></semantics></math></div> <p>where the top morphism is the inclusion of locally free <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>-modules.</p> <p>So <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>et</mi> <mn>1</mn></msubsup><mo stretchy="false">(</mo><mi>R</mi><mo>,</mo><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^1_{et}(R, GL_1)</annotation></semantics></math> is not equal to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mi>Pic</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_0 Pic(R)</annotation></semantics></math>, but it is off only by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>et</mi> <mn>0</mn></msubsup><mo stretchy="false">(</mo><mi>R</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>components</mi><mi>of</mi><mi>R</mi></mrow></msub><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">H^0_{et}(R, \mathbb{Z}) = \prod_{components of R} \mathbb{Z}</annotation></semantics></math>.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">Mod_R</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category">(infinity,1)-category</a> 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>-<a class="existingWikiWord" href="/nlab/show/module+spectra">modules</a>.</p> <p>There is a notion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">Mod_R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+%28infinity%2C1%29-category">enriched (infinity,1)-category</a>, 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>-linear <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories”.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Cat</mi> <mi>R</mi></msub><mo>≔</mo><msub><mi>Mod</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">Cat_R \coloneqq Mod_R</annotation></semantics></math>-modules in <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-categories">presentable (infinity,1)-categories</a>.</p> <p>Forming module <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories is then an <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-functor">(infinity,1)-functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Alg</mi> <mi>R</mi></msub><mover><mo>→</mo><mi>Mod</mi></mover><msub><mi>Cat</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex"> Alg_R \stackrel{Mod}{\to} Cat_R </annotation></semantics></math></div> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi><msub><mo>′</mo> <mi>R</mi></msub><mo>↪</mo><msub><mi>Cat</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">Cat'_R \hookrightarrow Cat_R</annotation></semantics></math> for the image of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mod</mi></mrow><annotation encoding="application/x-tex">Mod</annotation></semantics></math>. Then define the <a class="existingWikiWord" href="/nlab/show/Brauer+group">Brauer</a> <a class="existingWikiWord" href="/nlab/show/infinity-group">infinity-group</a> to be</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Br</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">(</mo><mi>Cat</mi><msub><mo>′</mo> <mi>R</mi></msub><msup><mo stretchy="false">)</mo> <mo>×</mo></msup></mrow><annotation encoding="application/x-tex"> Br(R) \coloneqq (Cat'_R)^\times </annotation></semantics></math></div> <p>One shows (<a href="#AntieauGepner12">Antieau-Gepner 12</a>) that this is exactly the Azumaya <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>-algebras modulo Morita equivalence.</p> <p><strong>Theorem</strong> (B. Antieau, D. Gepner)</p> <ol> <li> <p>For <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 connective <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">E_\infty</annotation></semantics></math> ring, any Azumaya <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>-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is étale locally trivial: there is an <a class="existingWikiWord" href="/nlab/show/etale+topology">etale cover</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">R \to S</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><msub><mo>∧</mo> <mi>R</mi></msub><mi>S</mi><mover><mo>→</mo><mrow><mi>Morita</mi><mo>≃</mo></mrow></mover><mi>S</mi></mrow><annotation encoding="application/x-tex">A \wedge_R S \stackrel{Morita \simeq}{\to} S</annotation></semantics></math>.</p> <p>(Think of this as saying that an Azumaya <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>-algebra is étale-locally a Matrix algebra, hence Morita-trivial: a “bundle of compact operators” presenting a (torsion) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL_1(R)</annotation></semantics></math>-2-bundle).</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Br</mi><mo>:</mo><msubsup><mi>CAlg</mi> <mi>R</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup><mo>→</mo><msub><mi>Gpd</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">Br : CAlg_R^{\geq 0} \to Gpd_\infty</annotation></semantics></math> is a sheaf for the <a class="existingWikiWord" href="/nlab/show/etale+cohomology">etale cohomology</a>.</p> </li> </ol> <p><strong>Corollary</strong></p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Br</mi></mrow><annotation encoding="application/x-tex">Br</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/connected+object+in+an+%28infinity%2C1%29-topos">connected</a>. Hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Br</mi><mo>≃</mo><msub><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>et</mi></msub><mi>Ω</mi><mi>Br</mi></mrow><annotation encoding="application/x-tex">Br \simeq \mathbf{B}_{et} \Omega Br </annotation></semantics></math>.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>Br</mi><mo>≃</mo><mi>Pic</mi></mrow><annotation encoding="application/x-tex">\Omega Br \simeq Pic</annotation></semantics></math>, hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Br</mi><mo>≃</mo><msub><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>et</mi></msub><mi>Pic</mi></mrow><annotation encoding="application/x-tex">Br \simeq \mathbf{B}_{et} Pic</annotation></semantics></math></p> </li> </ol> <p><a class="existingWikiWord" href="/nlab/show/Postnikov+tower">Postnikov tower</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL_1(R)</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>for</mi><mspace width="thickmathspace"></mspace><mi>n</mi><mo>></mo><mn>0</mn><mo>:</mo><msub><mi>π</mi> <mi>n</mi></msub><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>π</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex"> for\; n \gt 0: \pi_n GL_1(S) \simeq \pi_n </annotation></semantics></math></div> <p>hence for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">R \to S</annotation></semantics></math> étale</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mi>S</mi><mo>≃</mo><msub><mi>π</mi> <mi>n</mi></msub><mi>R</mi><msub><mo>⊗</mo> <mrow><msub><mi>π</mi> <mn>0</mn></msub><mi>R</mi></mrow></msub><msub><mi>π</mi> <mn>0</mn></msub><mi>S</mi></mrow><annotation encoding="application/x-tex"> \pi_n S \simeq \pi_n R \otimes_{\pi_0 R} \pi_0 S </annotation></semantics></math></div> <p>This is a <a class="existingWikiWord" href="/nlab/show/quasi-coherent+sheaf">quasi-coherent sheaf</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mi>R</mi></mrow><annotation encoding="application/x-tex">\pi_0 R</annotation></semantics></math> of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>N</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde N</annotation></semantics></math> (quasicoherent sheaf associated with a module), for <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> an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mi>R</mi></mrow><annotation encoding="application/x-tex">\pi_0 R</annotation></semantics></math>-module. By vanishing theorem of higher cohomology for quasicoherent sheaves</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>et</mi> <mn>1</mn></msubsup><mo stretchy="false">(</mo><msub><mi>π</mi> <mn>0</mn></msub><mi>R</mi><mo>,</mo><mover><mi>N</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>;</mo><mi>for</mi><mi>p</mi><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> H_{et}^1(\pi_0 R, \tilde N) = 0; for p \gt 0 </annotation></semantics></math></div> <p>For every <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-sheaf">(infinity,1)-sheaf</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/infinity-groups">infinity-groups</a>, there is a <a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>et</mi> <mi>p</mi></msubsup><mo stretchy="false">(</mo><msub><mi>π</mi> <mn>0</mn></msub><mi>R</mi><mo>;</mo><msub><mover><mi>π</mi><mo stretchy="false">˜</mo></mover> <mi>q</mi></msub><mi>G</mi><mo stretchy="false">)</mo><mo>⇒</mo><msub><mi>π</mi> <mrow><mi>q</mi><mo>−</mo><mi>p</mi></mrow></msub><mi>G</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H_{et}^p(\pi_0 R; \tilde \pi_q G) \Rightarrow \pi_{q-p} G(R) </annotation></semantics></math></div> <p>(the second argument on the left denotes the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>qth</mi></mrow><annotation encoding="application/x-tex">qth</annotation></semantics></math> Postnikov stage). From this one gets the following.</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>π</mi><mo stretchy="false">˜</mo></mover> <mn>0</mn></msub><mi>Br</mi><mo>≃</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\tilde \pi_0 Br \simeq *</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>π</mi><mo stretchy="false">˜</mo></mover> <mn>1</mn></msub><mi>Br</mi><mo>≃</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\tilde \pi_1 Br \simeq \mathbb{Z}</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>π</mi><mo stretchy="false">˜</mo></mover> <mn>2</mn></msub><mi>Br</mi><mo>≃</mo><msub><mover><mi>π</mi><mo stretchy="false">˜</mo></mover> <mn>1</mn></msub><mi>Pic</mi><mo>≃</mo><msub><mi>π</mi> <mn>0</mn></msub><msub><mi>GL</mi> <mn>1</mn></msub><mo>≃</mo><msub><mi>𝔾</mi> <mi>m</mi></msub></mrow><annotation encoding="application/x-tex">\tilde \pi_2 Br \simeq \tilde \pi_1 Pic \simeq \pi_0 GL_1 \simeq \mathbb{G}_m</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>π</mi><mo stretchy="false">˜</mo></mover> <mi>n</mi></msub><mi>Br</mi></mrow><annotation encoding="application/x-tex">\tilde \pi_n Br</annotation></semantics></math> is quasicoherent for <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 \gt 2</annotation></semantics></math>.</p> </li> </ul> <p>there is an <a class="existingWikiWord" href="/nlab/show/exact+sequence">exact sequence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><msubsup><mi>H</mi> <mi>et</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><msub><mi>π</mi> <mn>0</mn></msub><mi>R</mi><mo>,</mo><msub><mi>𝔾</mi> <mi>m</mi></msub><mo stretchy="false">)</mo><mo>→</mo><msub><mi>π</mi> <mn>0</mn></msub><mi>Br</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>→</mo><msubsup><mi>H</mi> <mi>et</mi> <mn>1</mn></msubsup><mo stretchy="false">(</mo><msub><mi>π</mi> <mn>0</mn></msub><mi>R</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0 \to H_{et}^2(\pi_0 R, \mathbb{G}_m) \to \pi_0 Br(R) \to H_{et}^1(\pi_0 R, \mathbb{Z}) \to 0 </annotation></semantics></math></div> <p>(notice the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Br</mi><mo stretchy="false">(</mo><msub><mi>π</mi> <mn>0</mn></msub><mi>R</mi><mo stretchy="false">)</mo><mo>↪</mo><msubsup><mi>H</mi> <mi>et</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><msub><mi>π</mi> <mn>0</mn></msub><mi>R</mi><mo>,</mo><msub><mi>𝔾</mi> <mi>m</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Br(\pi_0 R) \hookrightarrow H_{et}^2(\pi_0 R, \mathbb{G}_m)</annotation></semantics></math>)</p> <p>this is <a class="existingWikiWord" href="/nlab/show/split+exact+sequence">split exact</a> and so computes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mi>Br</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_0 Br(R)</annotation></semantics></math> for connective <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>.</p> <p>Now some more on the case that <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 not connective.</p> <p>Suppose there exists <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mover><mo>→</mo><mi>ϕ</mi></mover><mi>S</mi></mrow><annotation encoding="application/x-tex">R \stackrel{\phi}{\to} S</annotation></semantics></math> which is a faithful <a class="existingWikiWord" href="/nlab/show/Galois+extension">Galois extension</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/finite+group">finite group</a>.</p> <p><strong>Examples</strong></p> <ol> <li> <p>(real into complex <a class="existingWikiWord" href="/nlab/show/K-theory+spectrum">K-theory spectrum</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>KO</mi><mo>→</mo><mi>KU</mi></mrow><annotation encoding="application/x-tex">KO \to KU</annotation></semantics></math> (this is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tmf">tmf</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo><mi>tmf</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\to tmf(3)</annotation></semantics></math></p> </li> </ol> <p>Give <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">R \to S</annotation></semantics></math>, have a <a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Gl</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">/</mo><mi>S</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mi>fib</mi></mover><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Pic</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">/</mo><mi>S</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mi>fib</mi></mover><mi>Pic</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Pic</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Br</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">/</mo><mi>S</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mi>fib</mi></mover><mi>Br</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Br</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>→</mo><mi>⋯</mi></mrow><annotation encoding="application/x-tex"> Gl_1(R/S) \stackrel{fib}{\to} GL_1(R) \to GL_1(S) \to Pic(R/S) \stackrel{fib}{\to} Pic(R) \to Pic(S) \to Br(R/S) \stackrel{fib}{\to} Br(R) \to Br(S) \to \cdots </annotation></semantics></math></div> <p><strong>Theorem</strong> (descent theorems) (Tyler Lawson, David Gepner) Given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-Galois extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mover><mo>→</mo><mo>≃</mo></mover><msup><mi>S</mi> <mi>hG</mi></msup></mrow><annotation encoding="application/x-tex">R \stackrel{\simeq}{\to} S^{hG}</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/homotopy+fixed+points">homotopy fixed points</a>)</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>R</mi></msub><mover><mo>→</mo><mo>≃</mo></mover><msubsup><mi>Mod</mi> <mi>S</mi> <mi>hG</mi></msubsup></mrow><annotation encoding="application/x-tex">Mod_R \stackrel{\simeq}{\to} Mod_S^{hG}</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Alg</mi> <mi>R</mi></msub><mover><mo>→</mo><mo>≃</mo></mover><msubsup><mi>Alg</mi> <mi>S</mi> <mi>hG</mi></msubsup></mrow><annotation encoding="application/x-tex">Alg_R \stackrel{\simeq}{\to} Alg_S^{hG}</annotation></semantics></math></p> </li> </ol> <p>it follows that there is a homotopy fixed points spectral sequence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><msub><mi>π</mi> <mo>•</mo></msub><msup><mi>Σ</mi> <mi>n</mi></msup><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⇒</mo><msub><mi>π</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi></mrow></msub><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H^p(G, \pi_\bullet \Sigma^n GL_1(S)) \Rightarrow \pi_{-n} GL_1(S) </annotation></semantics></math></div> <p><strong>Conjecture</strong> The spectral sequence gives an Azumaya <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>KO</mi></mrow><annotation encoding="application/x-tex">KO</annotation></semantics></math>-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math> which is a nontrivial element in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Br</mi><mo stretchy="false">(</mo><mi>KO</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Br(KO)</annotation></semantics></math> but becomes trivial in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Br</mi><mo stretchy="false">(</mo><mi>KU</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Br(KU)</annotation></semantics></math>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Brauer+stack">Brauer stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/formal+Brauer+group">formal Brauer group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a>/<a class="existingWikiWord" href="/nlab/show/multiplicative+group">multiplicative group</a>, <a class="existingWikiWord" href="/nlab/show/Picard+group">Picard group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Azumaya+algebra">Azumaya algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brauer+%E2%88%9E-group">Brauer ∞-group</a></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> <p>Brauer groups are named after <a class="existingWikiWord" href="/nlab/show/Richard+Brauer">Richard Brauer</a>.</p> <p>Original discussion includes</p> <ul> <li id="Grothendieck64"> <p><a class="existingWikiWord" href="/nlab/show/Alexander+Grothendieck">Alexander Grothendieck</a>, <em>Le groupe de Brauer: II. Théories cohomologiques</em>. Séminaire Bourbaki, 9 (1964-1966), Exp. No. 297, 21 p. (<a href="http://www.numdam.org/item?id=SB_1964-1966__9__287_0">Numdam</a>)</p> </li> <li id="Grothendieck68"> <p><a class="existingWikiWord" href="/nlab/show/Alexandre+Grothendieck">Alexandre Grothendieck</a>, <em>Le groupe de Brauer</em>, <em>Dix exposés sur la cohomologie des schémas</em>, Masson and North-Holland, Paris and Amsterdam, (1968), pp. 46–66.</p> </li> </ul> <p>An introduction is in</p> <ul> <li>Pete Clark, <em>On the Brauer group</em> (2003) (<a href="http://math.uga.edu/~pete/trivial2003.pdf">pdf</a>)</li> </ul> <p>See also</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/John+Duskin">John Duskin</a>, <em>The Azumaya complex of a commutative ring</em>, in: Categorical algebra and its applications (Louvain-La-Neuve, 1987), 107–117, Lecture Notes in Math. <strong>1348</strong>, Springer 1988.</p> </li> <li id="Street"> <p><a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, <em>Descent</em>, Oberwolfach preprint (sec. 5, <em>Brauer groups</em>) <a href="http://www.math.mq.edu.au/~street/Descent.pdf">pdf</a>; <em>Some combinatorial aspects of descent theory</em>, Applied categorical structures <strong>12</strong> (2004) 537-576, <a href="http://arxiv.org/abs/math/0303175">math.CT/0303175</a> (sec. 12, <em>Brauer groups</em>)</p> </li> </ul> <p>The relation to <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a>/<a class="existingWikiWord" href="/nlab/show/etale+cohomology">etale cohomology</a> is discussed in</p> <ul id="Milne"> <li><a class="existingWikiWord" href="/nlab/show/James+Milne">James Milne</a>, <em>Étale cohomology</em>, Princeton Mathematical Series, vol. 33, Princeton University Press, Princeton, New Jersey (1980)</li> </ul> <ul id="Giraud"> <li><a class="existingWikiWord" href="/nlab/show/Jean+Giraud">Jean Giraud</a>, <em>Cohomologie non abelienne</em>, Die Grundlehren der mathematischen Wissenschaften, vol. 179, Springer-Verlag, Berlin, 1971.</li> </ul> <ul id="deJong"> <li> <p><a class="existingWikiWord" href="/nlab/show/Ofer+Gabber">Ofer Gabber</a>, <em>Some theorems on Azumaya algebras</em>, Ph. D. Thesis, Harvard University, 1978, Groupe de Brauer, Lecture Notes in Mathematics, vol. 844, Springer-Verlag, Berlin, 1981, pp. 129–209.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Aise+Johan+de+Jong">Aise Johan de Jong</a>, <em>A result of Gabber</em> (<a href="http://www.math.columbia.edu/~dejong/papers/2-gabber.pdf">pdf</a>)</p> </li> </ul> <ul id="Bertuccioni"> <li>Inta Bertuccioni, <em>Brauer groups and cohomology</em>, Archiv der Mathematik, vol. 84 Number 5 (2005)</li> </ul> <p>Brauer groups of <a class="existingWikiWord" href="/nlab/show/superalgebras">superalgebras</a> are discussed in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/C.+T.+C.+Wall">C. T. C. Wall</a>, <em>Graded Brauer groups</em>, J. Reine Angew. Math. 213 (1963/1964), 187-199.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pierre+Deligne">Pierre Deligne</a>, <em>Notes on spinors</em> in <em><a class="existingWikiWord" href="/nlab/show/Quantum+Fields+and+Strings">Quantum Fields and Strings</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Peter+Donovan">Peter Donovan</a>, <a class="existingWikiWord" href="/nlab/show/Max+Karoubi">Max Karoubi</a>, <em>Graded Brauer groups and K-theory with local coefficients</em>, Publications Math. IHES 38 (1970), 5-25 (<a href="http://www.math.jussieu.fr/~karoubi/Donavan.K.pdf">pdf</a>)</p> </li> </ul> <p>Refinement to <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a> and <a class="existingWikiWord" href="/nlab/show/Brauer+%E2%88%9E-groups">Brauer ∞-groups</a> is discussed in</p> <ul> <li id="Szymik11"> <p><a class="existingWikiWord" href="/nlab/show/Markus+Szymik">Markus Szymik</a>, <em>Brauer spaces for commutative rings and structured ring spectra</em> (<a href="http://arxiv.org/abs/1110.2956">arXiv:1110.2956</a>)</p> </li> <li id="BakerRichterSzymik12"> <p><a class="existingWikiWord" href="/nlab/show/Andrew+Baker">Andrew Baker</a>, <a class="existingWikiWord" href="/nlab/show/Birgit+Richter">Birgit Richter</a>, <a class="existingWikiWord" href="/nlab/show/Markus+Szymik">Markus Szymik</a>, <em>Brauer groups for commutative <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">\mathbb{S}</annotation></semantics></math>-algebras</em>, J. Pure Appl. Algebra 216 (2012) 2361–2376 (<a href="http://arxiv.org/abs/1005.5370">arXiv:1005.5370</a>)</p> </li> </ul> <p>Unification of all this in a theory of <a class="existingWikiWord" href="/nlab/show/%28infinity%2Cn%29-modules">(infinity,n)-modules</a> is in</p> <ul> <li id="Haugseng14"><a class="existingWikiWord" href="/nlab/show/Rune+Haugseng">Rune Haugseng</a>, <em>The higher Morita category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">E_n</annotation></semantics></math>-algebras</em> (<a href="http://arxiv.org/abs/1412.8459">arXiv:1412.8459</a>)</li> </ul> <p>The “bigger Brauer group” is discussed in</p> <ul> <li id="Taylor82"> <p>J. Taylor, <em>A bigger Brauer group</em> Pacific J. Math. 103 (1982), 163-203 (<a href="https://projecteuclid.org/euclid.pjm/1102724219">projecteuclid</a>)</p> </li> <li id="CaenepeelGrandjean98"> <p>S. Caenepeel, F. Grandjean, <em>A note on Taylor’s Brauer group</em>. Pacific J. Math. 186 (1998), 13-27</p> </li> <li id="HeinlothSchoeer08"> <p><a class="existingWikiWord" href="/nlab/show/Jochen+Heinloth">Jochen Heinloth</a>, Stefan Schröer, <em>The bigger Brauer group and twisted sheaves</em> (<a href="http://arxiv.org/abs/0803.3563">arXiv:0803.3563</a>)</p> </li> </ul> <p>See also</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jochen+Heinloth">Jochen Heinloth</a>, <a class="existingWikiWord" href="/nlab/show/Marc+Levine">Marc Levine</a>, Stefan Scröer, <em>Forschungsseminar: Brauer groups and Artin stack</em>, 07 (<a href="https://www.uni-due.de/~mat903/sem/brauer.pdf">pdf</a>)</li> </ul> <p>The observation that passing to <a class="existingWikiWord" href="/nlab/show/derived+algebraic+geometry">derived algebraic geometry</a> makes also the non-torsion elements in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>et</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><msub><mi>𝔾</mi> <mi>m</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^2_{et}(-,\mathbb{G}_m)</annotation></semantics></math> be represented by (derived) <a class="existingWikiWord" href="/nlab/show/Azumaya+algebras">Azumaya algebras</a> is due to</p> <ul> <li id="Toen10"><a class="existingWikiWord" href="/nlab/show/Bertrand+To%C3%ABn">Bertrand Toën</a>, <em>Derived Azumaya algebras and generators for twisted derived categories</em> (<a href="http://arxiv.org/abs/1002.2599">arXiv:1002.2599</a>)</li> </ul> <p>Related MO discussion includes</p> <ul> <li><a href="http://mathoverflow.net/questions/87345/brauer-groups-and-k-theory">Brauer groups and K-theory</a></li> </ul> <p>Systematic discussion of Brauer groups in <a class="existingWikiWord" href="/nlab/show/derived+algebraic+geometry">derived algebraic geometry</a> is in</p> <ul> <li id="AntieauGepner12"><a class="existingWikiWord" href="/nlab/show/Benjamin+Antieau">Benjamin Antieau</a>, <a class="existingWikiWord" href="/nlab/show/David+Gepner">David Gepner</a>, <em>Brauer groups and étale cohomology in derived algebraic geometry</em>, Geom. Topol. 18 (2014) 1149-1244 (<a href="http://arxiv.org/abs/1210.0290">arXiv:1210.0290</a>)</li> </ul> <p>For the Brauer-Picard 2-group of a tensor category, see</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Alexei+Davydov">Alexei Davydov</a>, <a class="existingWikiWord" href="/nlab/show/Dmitri+Nikshych">Dmitri Nikshych</a>, <em>Braided Picard groups and graded extensions of braided tensor categories</em>, (<a href="https://arxiv.org/abs/2006.08022">arXiv:2006.08022</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 6, 2023 at 04:30:54. See the <a href="/nlab/history/Brauer+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/Brauer+group" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/3905/#Item_12">Discuss</a><span class="backintime"><a href="/nlab/revision/Brauer+group/32" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/Brauer+group" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/Brauer+group" accesskey="S" class="navlink" id="history" rel="nofollow">History (32 revisions)</a> <a href="/nlab/show/Brauer+group/cite" style="color: black">Cite</a> <a href="/nlab/print/Brauer+group" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/Brauer+group" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>