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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{Brauer group} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra} [[!include higher algebra - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{RelationToCatsOfModules}{Relation to categories of modules}\dotfill \pageref*{RelationToCatsOfModules} \linebreak \noindent\hyperlink{RelationToEtaleCohomology}{Relation to \'e{}tale cohomology}\dotfill \pageref*{RelationToEtaleCohomology} \linebreak \noindent\hyperlink{RelationToDerivedEtaleCohomology}{Relation to derived \'e{}tale cohomology}\dotfill \pageref*{RelationToDerivedEtaleCohomology} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} For $R$ a [[ring]], the \emph{Brauer group} $Br(R)$ is the [[group]] of [[Morita equivalence]] classes of [[Azumaya algebras]] over $R$. \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{RelationToCatsOfModules}{}\subsubsection*{{Relation to categories of modules}}\label{RelationToCatsOfModules} \begin{defn} \label{}\hypertarget{}{} For $R$ a commutative [[ring]], let $Alg_R$ or $2Vect_R$ (see at [[2-vector space]]/[[2-module]]) be the [[2-category]] whose \begin{itemize}% \item [[objects]] are $R$-[[associative algebra|algebras]]; \item [[morphisms]] are [[bimodules]] over $R$-algebras; \item [[2-morphisms]] are bimodule homomorphisms. \end{itemize} \end{defn} \begin{remark} \label{}\hypertarget{}{} This may be understood as the 2-category of (generalized) [[2-vector bundles]] over $Spec R$, the [[Isbell duality|formally dual]] [[space]] whose [[function algebra]] is $R$. This is a [[braided monoidal category|braided monoidal 2-category]]. \end{remark} \begin{defn} \label{}\hypertarget{}{} Let \begin{displaymath} \mathbf{Br}(R) \coloneqq Core(Alg_R) \end{displaymath} be its [[Picard 3-group]], hence the maximal [[infinity-group|3-group]] inside (which is hence a [[braided 3-group]]), the [[core]] on the invertible objects, hence the [[2-groupoid]] whose \begin{itemize}% \item [[objects]] are algebras which are invertible up to [[Morita equivalence]] under tensor product; \item [[morphisms]] are [[Morita equivalences]]; \item [[2-morphisms]] are invertible bimodule homomorphisms. \end{itemize} \end{defn} \begin{remark} \label{}\hypertarget{}{} This may be understood as the 2-groupoid of (generalized) [[line 2-bundles]] over $Spec R$ (for instance [[holomorphic line 2-bundles]] in the case of [[higher complex analytic geometry]]), inside that of all [[2-vector bundles]]. \end{remark} \begin{prop} \label{}\hypertarget{}{} The [[homotopy groups]] of $\mathbf{Br}(R)$ are the following: \begin{itemize}% \item $\pi_0(\mathbf{Br}(R))$ is the Brauer group of $R$; \item $\pi_1(\mathbf{Br}(R))$ is the [[Picard group]] of $R$; \item $\pi_2(\mathbf{Br}(R))$ is the [[group of units]] of $R$. \end{itemize} \end{prop} See for instance (\hyperlink{Street}{Street}). \begin{example} \label{}\hypertarget{}{} Analogous statements hold for (non-commutative) [[superalgebras]], hence for $\mathbb{Z}_2$-[[graded algebras]]. See at \emph{\href{super+algebra#Picard2Groupoid}{superalgebra -- Picard 3-group, Brauer group}}. \end{example} \hypertarget{RelationToEtaleCohomology}{}\subsubsection*{{Relation to \'e{}tale cohomology}}\label{RelationToEtaleCohomology} The Brauer group of a [[ring]] $R$ is a [[torsion]] subgroup of the second [[etale cohomology]] group of $Spec R$ with values in the [[multiplicative group]] $\mathbb{G}_m$ \begin{displaymath} Br(X) \hookrightarrow H^2_{et}(X, \mathbb{G}_m) \,. \end{displaymath} This was first stated in (\hyperlink{Grothendieck68}{Grothendieck 68}) (see also \hyperlink{Grothendieck64}{Grothendieck 64, prop. 1.4} and see at \emph{\href{algebraic+line+n-bundle#Properties}{algebraic line n-bundle -- Properties}}). Review discussion is in (\hyperlink{Milne}{Milne, chapter IV}). A detailed discussion in the context of [[nonabelian cohomology]] is in (\hyperlink{Giraud}{Giraud}). A theorem stating conditions under which the Brauer group is precisely the [[torsion]] subgroup of $H^2_{et}(X, \mathbb{G}_m)$ is due to (\hyperlink{Gabber}{Gabber}), see also the review in (\hyperlink{deJong}{de Jong}). For more details and more literature on this see (\hyperlink{Bertuccioni}{Bertuccioni}). This fits into the following pattern \begin{itemize}% \item $H^0_{et}(R, \mathbb{G}_m) = R^\times$ ([[group of units]]) \item $H^1_{et}(R, \mathbb{G}_m) = Pic(R)$ ([[Picard group]]: iso classes of invertible $R$-modules) \item $H^2_{et}(R, \mathbb{G}_m)_{tor} = Br(R)$ ([[Brauer group]]: [[Morita equivalence]] classes of [[Azumaya algebras]] over $R$) (the torsion equivalence classes of the [[Brauer stack]]) \end{itemize} It is therefore natural to regard all of $H^2_{et}(R, \mathbb{G}_m)$ as the ``actual'' Brauer group. This has been called the ``[[bigger Brauer group]]'' (\hyperlink{Taylor82}{Taylor 82}, \hyperlink{CaenepeelGrandjean98}{Caenepeel-Grandjean 98}, \hyperlink{HeinlothSchoeer08}{Heinloth-Sch\"o{}er 08}). The bigger Brauer group has actually traditionally been implicit already in the term ``[[formal Brauer group]]'', which is really the [[formal geometry]]-version of the bigger Brauer group. \hypertarget{RelationToDerivedEtaleCohomology}{}\subsubsection*{{Relation to derived \'e{}tale cohomology}}\label{RelationToDerivedEtaleCohomology} More generally, this works for $R$ a (connective) [[E-infinity ring]] (the following is due to \hyperlink{AntieauGepner12}{Antieau-Gepner 12}, see \hyperlink{Haugseng14}{Haugseng 14} for more). Let $GL_1(R)$ be its [[infinity-group of units]]. If $R$ is [[connective spectrum|connective]], then the first [[Postnikov tower|Postnikov stage]] of the [[Picard group|Picard]] [[infinity-groupoid]] \begin{displaymath} Pic(R) \coloneqq Mod(R)^\times \end{displaymath} is \begin{displaymath} \itexarray{ \mathbf{B}_{et} GL_1(-) &\to& Pic(-) \\ && \downarrow \\ && \mathbb{Z} } \,, \end{displaymath} where the top morphism is the inclusion of locally free $R$-modules. So $H^1_{et}(R, GL_1)$ is not equal to $\pi_0 Pic(R)$, but it is off only by $H^0_{et}(R, \mathbb{Z}) = \prod_{components of R} \mathbb{Z}$. Let $Mod_R$ be the [[(infinity,1)-category]] of $R$-[[module spectra|modules]]. There is a notion of $Mod_R$-[[enriched (infinity,1)-category]], of ``$R$-linear $(\infty,1)$-categories''. $Cat_R \coloneqq Mod_R$-modules in [[presentable (infinity,1)-categories]]. Forming module $(\infty,1)$-categories is then an [[(infinity,1)-functor]] \begin{displaymath} Alg_R \stackrel{Mod}{\to} Cat_R \end{displaymath} Write $Cat'_R \hookrightarrow Cat_R$ for the image of $Mod$. Then define the [[Brauer group|Brauer]] [[infinity-group]] to be \begin{displaymath} Br(R) \coloneqq (Cat'_R)^\times \end{displaymath} One shows (\hyperlink{AntieauGepner12}{Antieau-Gepner 12}) that this is exactly the Azumaya $R$-algebras modulo Morita equivalence. \textbf{Theorem} (B. Antieau, D. Gepner) \begin{enumerate}% \item For $R$ a connective $E_\infty$ ring, any Azumaya $R$-algebra $A$ is \'e{}tale locally trivial: there is an [[etale topology|etale cover]] $R \to S$ such that $A \wedge_R S \stackrel{Morita \simeq}{\to} S$. (Think of this as saying that an Azumaya $R$-algebra is \'e{}tale-locally a Matrix algebra, hence Morita-trivial: a ``bundle of compact operators'' presenting a (torsion) $GL_1(R)$-2-bundle). \item $Br : CAlg_R^{\geq 0} \to Gpd_\infty$ is a sheaf for the [[etale cohomology]]. \end{enumerate} \textbf{Corollary} \begin{enumerate}% \item $Br$ is [[connected object in an (infinity,1)-topos|connected]]. Hence $Br \simeq \mathbf{B}_{et} \Omega Br$. \item $\Omega Br \simeq Pic$, hence $Br \simeq \mathbf{B}_{et} Pic$ \end{enumerate} [[Postnikov tower]] for $GL_1(R)$: \begin{displaymath} for\; n \gt 0: \pi_n GL_1(S) \simeq \pi_n \end{displaymath} hence for $R \to S$ \'e{}tale \begin{displaymath} \pi_n S \simeq \pi_n R \otimes_{\pi_0 R} \pi_0 S \end{displaymath} This is a [[quasi-coherent sheaf]] on $\pi_0 R$ of the form $\tilde N$ (quasicoherent sheaf associated with a module), for $N$ an $\pi_0 R$-module. By vanishing theorem of higher cohomology for quasicoherent sheaves \begin{displaymath} H_{et}^1(\pi_0 R, \tilde N) = 0; for p \gt 0 \end{displaymath} For every [[(infinity,1)-sheaf]] $G$ of [[infinity-groups]], there is a [[spectral sequence]] \begin{displaymath} H_{et}^p(\pi_0 R; \tilde \pi_q G) \Rightarrow \pi_{q-p} G(R) \end{displaymath} (the second argument on the left denotes the $qth$ Postnikov stage). From this one gets the following. \begin{itemize}% \item $\tilde \pi_0 Br \simeq *$ \item $\tilde \pi_1 Br \simeq \mathbb{Z}$; \item $\tilde \pi_2 Br \simeq \tilde \pi_1 Pic \simeq \pi_0 GL_1 \simeq \mathbb{G}_m$ \item $\tilde \pi_n Br$ is quasicoherent for $n \gt 2$. \end{itemize} there is an [[exact sequence]] \begin{displaymath} 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 \end{displaymath} (notice the inclusion $Br(\pi_0 R) \hookrightarrow H_{et}^2(\pi_0 R, \mathbb{G}_m)$) this is [[split exact sequence|split exact]] and so computes $\pi_0 Br(R)$ for connective $R$. Now some more on the case that $R$ is not connective. Suppose there exists $R \stackrel{\phi}{\to} S$ which is a faithful [[Galois extension]] for $G$ a [[finite group]]. \textbf{Examples} \begin{enumerate}% \item (real into complex [[K-theory spectrum]]) $KO \to KU$ (this is $\mathbb{Z}_2$) \item [[tmf]] $\to tmf(3)$ \end{enumerate} Give $R \to S$, have a [[fiber sequence]] \begin{displaymath} 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 \end{displaymath} \textbf{Theorem} (descent theorems) (Tyler Lawson, David Gepner) Given $G$-Galois extension $R \stackrel{\simeq}{\to} S^{hG}$ ([[homotopy fixed points]]) \begin{enumerate}% \item $Mod_R \stackrel{\simeq}{\to} Mod_S^{hG}$ \item $Alg_R \stackrel{\simeq}{\to} Alg_S^{hG}$ \end{enumerate} it follows that there is a homotopy fixed points spectral sequence \begin{displaymath} H^p(G, \pi_\bullet \Sigma^n GL_1(S)) \Rightarrow \pi_{-n} GL_1(S) \end{displaymath} \textbf{Conjecture} The spectral sequence gives an Azumaya $KO$-algebra $Q$ which is a nontrivial element in $Br(KO)$ but becomes trivial in $Br(KU)$. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[Brauer stack]] \item [[formal Brauer group]] \item [[group of units]]/[[multiplicative group]], [[Picard group]] \item [[Azumaya algebra]] \item [[Brauer ∞-group]] \end{itemize} [[!include moduli of higher lines -- table]] \hypertarget{references}{}\subsection*{{References}}\label{references} Brauer groups are named after [[Richard Brauer]]. Original discussion includes \begin{itemize}% \item [[Alexander Grothendieck]], \emph{Le groupe de Brauer : II. Th\'e{}ories cohomologiques}. S\'e{}minaire Bourbaki, 9 (1964-1966), Exp. No. 297, 21 p. (\href{http://www.numdam.org/item?id=SB_1964-1966__9__287_0}{Numdam}) \item [[Alexandre Grothendieck]], \emph{Le groupe de Brauer}, Dix expos\'e{}s sur la cohomologie des sch\'e{}mas\_, Masson and North-Holland, Paris and Amsterdam, (1968), pp. 46--66. \end{itemize} An introduction is in \begin{itemize}% \item Pete Clark, \emph{On the Brauer group} (2003) (\href{http://math.uga.edu/~pete/trivial2003.pdf}{pdf}) \end{itemize} See also \begin{itemize}% \item [[John Duskin]], \emph{The Azumaya complex of a commutative ring}, in: Categorical algebra and its applications (Louvain-La-Neuve, 1987), 107--117, Lecture Notes in Math. \textbf{1348}, Springer 1988. \item [[Ross Street]], \emph{Descent}, Oberwolfach preprint (sec. 6, \emph{Brauer groups}) \href{http://www.math.mq.edu.au/~street/Descent.pdf}{pdf}; \emph{Some combinatorial aspects of descent theory}, Applied categorical structures \textbf{12} (2004) 537-576, \href{http://arxiv.org/abs/math/0303175}{math.CT/0303175} (sec. 12, \emph{Brauer groups}) \end{itemize} The relation to [[cohomology]]/[[etale cohomology]] is discussed in \begin{itemize}% \item [[James Milne]], \emph{\'E{}tale cohomology}, Princeton Mathematical Series, vol. 33, Princeton University Press, Princeton, New Jersey (1980) \end{itemize} \begin{itemize}% \item [[Jean Giraud]], \emph{Cohomologie non abelienne}, Die Grundlehren der mathematischen Wissenschaften, vol. 179, Springer- Verlag, Berlin, 1971. \end{itemize} \begin{itemize}% \item [[Ofer Gabber]], \emph{Some theorems on Azumaya algebras}, Ph. D. Thesis, Harvard University, 1978, Groupe de Brauer, Lecture Notes in Mathematics, vol. 844, Springer-Verlag, Berlin, 1981, pp. 129--209. \item [[Aise Johan de Jong]], \emph{A result of Gabber} (\href{http://www.math.columbia.edu/~dejong/papers/2-gabber.pdf}{pdf}) \end{itemize} \begin{itemize}% \item Inta Bertuccioni, \emph{Brauer groups and cohomology}, Archiv der Mathematik, vol. 84 Number 5 (2005) \end{itemize} Brauer groups of [[superalgebras]] are discussed in \begin{itemize}% \item [[C. T. C. Wall]], \emph{Graded Brauer groups}, J. Reine Angew. Math. 213 (1963/1964), 187-199. \item [[Pierre Deligne]], \emph{Notes on spinors} in \emph{[[Quantum Fields and Strings]]} \item [[Peter Donovan]], [[Max Karoubi]], \emph{Graded Brauer groups and K-theory with local coefficients}, Publications Math. IHES 38 (1970), 5-25 (\href{http://www.math.jussieu.fr/~karoubi/Donavan.K.pdf}{pdf}) \end{itemize} Refinement to [[stable homotopy theory]] and [[Brauer ∞-groups]] is discussed in \begin{itemize}% \item [[Markus Szymik]], \emph{Brauer spaces for commutative rings and structured ring spectra} (\href{http://arxiv.org/abs/1110.2956}{arXiv:1110.2956}) \item [[Andrew Baker]], [[Birgit Richter]], [[Markus Szymik]], \emph{Brauer groups for commutative $\mathbb{S}$-algebras}, J. Pure Appl. Algebra 216 (2012) 2361--2376 (\href{http://arxiv.org/abs/1005.5370}{arXiv:1005.5370}) \end{itemize} Unification of all this in a theory of [[(infinity,n)-modules]] is in \begin{itemize}% \item [[Rune Haugseng]], \emph{The higher Morita category of $E_n$-algebras} (\href{http://arxiv.org/abs/1412.8459}{arXiv:1412.8459}) \end{itemize} The ``bigger Brauer group'' is discussed in \begin{itemize}% \item J. Taylor, \emph{A bigger Brauer group} Pacic J. Math. 103 (1982), 163-203 (\href{https://projecteuclid.org/euclid.pjm/1102724219}{projecteuclid}) \item S. Caenepeel, F. Grandjean, \emph{A note on Taylor's Brauer group}. Pacific J. Math. 186 (1998), 13-27 \item [[Jochen Heinloth]], Stefan Schr\"o{}er, \emph{The bigger Brauer group and twisted sheaves} (\href{http://arxiv.org/abs/0803.3563}{arXiv:0803.3563}) \end{itemize} See also \begin{itemize}% \item [[Jochen Heinloth]], [[Marc Levine]], Stefan Scr\"o{}er, \emph{Forschungsseminar: Brauer groups and Artin stack}, 07 (\href{https://www.uni-due.de/~mat903/sem/brauer.pdf}{pdf}) \end{itemize} The observation that passing to [[derived algebraic geometry]] makes also the non-torsion elements in $H^2_{et}(-,\mathbb{G}_m)$ be represented by (derived) [[Azumaya algebras]] is due to \begin{itemize}% \item [[Bertrand Toën]], \emph{Derived Azumaya algebras and generators for twisted derived categories} (\href{http://arxiv.org/abs/1002.2599}{arXiv:1002.2599}) \end{itemize} Related MO discussion includes \begin{itemize}% \item \href{http://mathoverflow.net/questions/87345/brauer-groups-and-k-theory}{Brauer groups and K-theory} \end{itemize} Systematic discussion of Brauer groups in [[derived algebraic geometry]] is in \begin{itemize}% \item [[Benjamin Antieau]], [[David Gepner]], \emph{Brauer groups and \'e{}tale cohomology in derived algebraic geometry}, Geom. Topol. 18 (2014) 1149-1244 (\href{http://arxiv.org/abs/1210.0290}{arXiv:1210.0290}) \end{itemize} [[!redirects Brauer groups]] [[!redirects bigger Brauer group]] [[!redirects bigger Brauer groups]] \end{document}