symmetric monoidal (∞,1)-category of spectra
Given a commutative unital ring $R$, an Azumaya $R$-algebra is a (noncommutative in general) $R$-algebra $A$ which is finitely generated faithful 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.
More generally, Grothendieck defines an 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 $\mathcal{A}\otimes_{\mathcal{O}_X} \mathbf{End}(\mathcal{E}) \cong \mathcal{A}\otimes_{\mathcal{O}_X}\mathbf{End}(\mathcal{F})$ 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 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.
For $R$ a ring and $H^n_{et}(-,-)$ the etale cohomology, $\mathbb{G}_m$ the multiplicative group of the affine line; then
$H^0_{et}(R, \mathbb{G}_m) = R^\times$ (group of units)
$H^1_{et}(R, \mathbb{G}_m) = Pic(R)$ (Picard group: iso classes of invertible $R$-modules)
$H^2_{et}(R, \mathbb{G}_m)_{tor} = Br(R)$ (Brauer group Morita classes of Azumaya $R$-algebras)
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, then the first Postikov stage of the Picard infinity-groupoid
is
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 $R$-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
Write $Cat'_R \hookrightarrow Car_R$ for the image of $Mod$. Then define the Brauer infinity-group to be
One shows (Antieau-Gepner) that this is exactly the Azumaya $R$-algebras modulo Morita equivalence.
Theorem (B. Antieau, D. Gepner)
For $R$ a connective $E_\infty$ ring, any Azumaya $R$-algebra $A$ is étale locally trivial: there is an 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 étale-locally a Matric algebra, hence Morita-trivial: a “bundle of compact operators” presenting a (torsion) $GL_1(R)$-2-bundle).
$Br : CAlg_R^{\geq 0} \to Gpd_\infty$ is a sheaf for the etale cohomology.
Corollary
$Br$ is connected. Hence $Br \simeq \mathbf{B}_{et} \Omega Br$.
$\Omega Br \simeq Pic$, hence $Br \simeq \mathbf{B}_{et} Pic$
Postnikov tower for $GL_1(R)$:
hence for $R \to S$ étale
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
For every (infinity,1)-sheaf $G$ of infinity-groups, there is a spectral sequence
(the second argument on the left denotes the $qth$ Postnikov stage). From this one gets the following.
$\tilde \pi_0 Br \simeq *$
$\tilde \pi_1 Br \simeq \mathbb{Z}$;
$\tilde \pi_2 Br \simeq \tilde \pi_1 Pic \simeq \pi_0 GL_1 \simeq \mathbb{G}_m$
$\tilde \pi_n Br$ is quasicoherent for $n \gt 2$.
there is an exact sequence
(notice the inclusion $Br(\pi_0 R) \hookrightarrow H_{et}^2(\pi_0 R, \mathbb{G}_m)$)
this is 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.
Examples
(real into complex K-theory spectrum) $KO \to KU$ (this is $\mathbb{Z}_2$)
tmf $\to tmf(3)$
Give $R \to S$, have a fiber sequence
Theorem (descent theorems) (Tyler Lawson, David Gepner) Given $G$-Galois extension $R \stackrel{\simeq}{\to} S^{hG}$ (homotopy fixed points)
$Mod_R \stackrel{\simeq}{\to} Mod_S^{hG}$
$Alg_R \stackrel{\simeq}{\to} Alg_S^{hG}$
it follows that there is a homotopy fixed points spectral sequence
Conjecture The spectral sequence gives an Azumaya $KO$-algebra $Q$ which is a nontrivial element in $Br(KO)$ but becomes trivial in $Br(KU)$.
G. Cortiñas, Charles Weibel, Homology of Azumaya algebras, Proc. AMS 121, 1, pp. 1994 (jstor)
John Duskin, The Azumaya complex of a commutative ring, in Categorical Algebra and its Appl., Lec. Notes in Math. 1348 (1988) doi:10.1007/BFb0081352
Alexander Grothendieck, Le groupe de Brauer I, II, III, in Dix exposes sur la cohomologie des schemas (I: Algèbres d’Azumaya et interprétations diverses) North-Holland Pub. Co., Amsterdam (1969)
Max Karoubi, Peter Donovan, Graded Brauer groups and $K$-theory with local coefficients (pdf)
M-A. Knus, M. Ojanguren, Théorie de la descente et algèbres d’Azumaya, Lec. Notes in Math. 389, Springer 1974, doi:10.1007/BFb0057799, MR0417149
J. Milne, Étale cohomology, Princeton Univ. Press
Ross Street, Descent, Oberwolfach preprint (sec. 6, Brower groups) pdf; Some combinatorial aspects of descent theory, Applied categorical structures 12 (2004) 537-576, math.CT/0303175 (sec. 12, Brower groups)
Enrico Vitale, A Picard-Brauer exact sequence of categorical groups, pdf
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
See also
Category Cafe 2006: Picard and Brauer 2-groups