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\section*{Azumaya algebra}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{traditional}{Traditional}\dotfill \pageref*{traditional} \linebreak
\noindent\hyperlink{InTermsOfEtaleCohomology}{In terms of (derived) \'e{}tale cohomology}\dotfill \pageref*{InTermsOfEtaleCohomology} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{traditional}{}\subsubsection*{{Traditional}}\label{traditional}

Given a [[commutative unital ring]] $R$, an \textbf{Azumaya $R$-algebra} is a (noncommutative in general) $R$-[[associative algebra|algebra]] $A$ which is finitely generated faithful [[projective object|projective]] as an $R$-[[module]] and the canonical morphism $A\otimes_R A^{op}\to End_R(A)$ is an [[isomorphism]]. This definition extends the notion of a [[central simple algebra]] over a [[field]]; partly by this reason, Azumaya algebras are sometimes called \textbf{central separable $R$-algebras}.

More generally, [[Grothendieck]] defines an \textbf{Azumaya algebra} over a [[scheme]] $X$ as a [[sheaf]] $\mathcal{A}$ of $\mathcal{O}_X$-algebras such that for each point $x\in X$, the corresponding [[stalk]] $\mathcal{A}_x$ is an Azumaya $\mathcal{O}_{X,x}$-algebra.

The [[Brauer group]] $Br(X)$ classifies Azumaya algebras over $X$ up to a suitably defined equivalence relation: $\mathcal{A}\sim\mathcal{B}$ if $\mathcal{A}\otimes_{\mathcal{O}_X} \mathbf{End}(\mathcal{E}) \cong \mathcal{B}\otimes_{\mathcal{O}_X}\mathbf{End}(\mathcal{F})$ for some locally free sheaves of $\mathcal{O}_X$-modules $\mathcal{E}$ and $\mathcal{F}$ of finite rank. The group operation of $Br(X)$ is induced by the tensor product. The Brauer group can be reexpressed in terms of second [[nonabelian cohomology]]; indeed a sheaf of Azumaya algebras over $X$ determines an $\mathcal{O}_X^*$-[[gerbe]] (or $U(1)$-gerbe in the [[manifold]] context).

Brauer groups and Azumaya algebras are closely related to [[Morita theory]] and they make sense in the context of algebras and bimodules in the context of [[braided monoidal category|braided monoidal categories]]. [[Karoubi K-theory]] involves an element in a Brauer group and in the original Karoubi--Donovan paper is related to a twisting with a ``local system'' which involves Azumaya algebras.

\hypertarget{InTermsOfEtaleCohomology}{}\subsubsection*{{In terms of (derived) \'e{}tale cohomology}}\label{InTermsOfEtaleCohomology}

For $R$ a [[ring]] and $H^n_{et}(-,-)$ the [[etale cohomology]], $\mathbb{G}_m$ the [[multiplicative group]] of the [[affine line]]; then

\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 classes of Azumaya $R$-algebras)



\end{itemize}
More generally, this works for $R$ a (connective) [[E-infinity ring]] (the following is due to [[Benjamin Antieau]] and [[David Gepner]]).

Let $GL_1(R)$ be its [[infinity-group of units]]. If $R$ is [[connective spectrum|connective]], then the first [[Postnikov tower|Postikov 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''.

$Cat_R \coloneqq Mod_R$-modiles 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 $Cat'_R \hookrightarrow Car_R$ 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 (Antieau-Gepner) 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 Matric 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 [[group of units]], [[Picard group]], [[Brauer group]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item \href{http://mate.dm.uba.ar/~gcorti}{G. Corti\~n{}as}, [[Charles Weibel]], \emph{Homology of Azumaya algebras}, Proc. AMS \textbf{121}, 1, pp. 1994 (\href{http://www.jstor.org/stable/2160364}{jstor})


\item [[John Duskin]], \emph{The Azumaya complex of a commutative ring}, in Categorical Algebra and its Appl., Lec. Notes in Math. 1348 (1988) \href{http://dx.doi.org/10.1007/BFb0081352}{doi:10.1007/BFb0081352}


\item [[Alexander Grothendieck]], \emph{Le groupe de Brauer I, II, III}, in Dix exposes sur la cohomologie des schemas (I: Alg\`e{}bres d'Azumaya et interpr\'e{}tations diverses) North-Holland Pub. Co., Amsterdam (1969)


\item [[Max Karoubi]], [[Peter Donovan]], \emph{Graded Brauer groups and $K$-theory with local coefficients} (\href{http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1970__38_/PMIHES_1970__38__5_0/PMIHES_1970__38__5_0.pdf}{pdf})


\item M-A. Knus, M. Ojanguren, \emph{Th\'e{}orie de la descente et alg\`e{}bres d'Azumaya}, Lec. Notes in Math. \textbf{389}, Springer 1974, \href{http://dx.doi.org/doi:10.1007/BFb0057799}{doi:10.1007/BFb0057799}, MR0417149


\item J. Milne, \emph{\'E{}tale cohomology}, Princeton Univ. Press


\item [[Ross Street]], \emph{Descent}, Oberwolfach preprint (sec. 6, Brower 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, Brower groups)


\item [[Enrico Vitale]], \emph{A Picard-Brauer exact sequence of categorical groups}, \href{http://www.math.ucl.ac.be/membres/vitale/cat-gruppi2.pdf}{pdf}



\end{itemize}
The observation that passing to [[derived algebraic geometry]] makes also the non-torsion elements in the ``[[bigger Brauer group]]'' $H^2_{et}(-,\mathbb{G}_m)$ 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}
The comparison of the Artin's theorem on characterization of Azumaya algebras and Tomiyama-Takesaki's theorem on $n$-[[homogeneous C\emph{-algebra]]s is in chapter 9 of}

\begin{itemize}%
\item Edward Formanek, \emph{Noncommutative invariant theory}, in: Group actions on rings (Brunswick, Maine, 1984), 87--119, Contemp. Math. 43, Amer. Math. Soc. 1985 \href{http://dx.doi.org/10.1090/conm/043}{doi}

\end{itemize}
See also

\begin{itemize}%
\item \href{http://en.wikipedia.org/wiki/Azumaya_algebra}{wikipedia page}


\item Category Cafe 2006: \href{http://golem.ph.utexas.edu/string/archives/000786.html}{Picard and Brauer 2-groups}



\end{itemize}
[[!redirects Azumaya algebras]] [[!redirects central separable algebra]]



\end{document}