symmetric monoidal (∞,1)-category of spectra
For $R$ a ring, the Brauer group $Br(R)$ is the group of Morita equivalence classes of Azumaya algebras over $R$.
For $R$ a commutative ring, let $Alg_R$ or $2Vect_R$ (see at 2-vector space/2-module) be the 2-category whose
2-morphisms are bimodule homomorphisms.
This may be understood as the 2-category of (generalized) 2-vector bundles over $Spec R$, the formally dual space whose function algebra is $R$. This is a braided monoidal 2-category.
Let
be its Picard 3-group, hence the maximal 3-group inside (which is hence a braided 3-group), the core on the invertible objects, hence the 2-groupoid whose
objects are algebras which are invertible up to Morita equivalence under tensor product;
2-morphisms are invertible bimodule homomorphisms.
This may be understood as the 2-groupoid of (generalized) line 2-bundles over $Spec R$, inside that of all 2-vector bundles.
The homotopy groups of $\mathbf{Br}(R)$ are the following:
$\pi_0(\mathbf{Br}(R))$ is the Brauer group of $R$;
$\pi_1(\mathbf{Br}(R))$ is the Picard group of $R$;
$\pi_2(\mathbf{Br}(R))$ is the group of units of $R$.
See for instance (Street).
Analgous statements hold for (non-commutative) superalgebras, hence for $\mathbb{Z}_2$-graded algebras. See at superalgebra – Picard 3-group, Brauer group.
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 $\mathbb{G}_m$
This was first stated in (Grothendieck 68), a discussion is in chapter IV of (Milne). A detailed discussion in the context of nonabelian cohomology is in (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 (Gabber), see also the review in (de Jong). For more details and more literature on this see (Bertuccioni).
This fits into the following pattern
$H^0_{et}(R, \mathbb{G}_m) = R^\times$ (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 equivalence classes of Azumaya algebras over $R$)
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 $R \to S$ eétale
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 $Br(KU)$.
Brauer groups are named after Richard Brauer.
An introduction is in
Pete Clark, On the Brauer group (2003) (pdf)
Alexandre Grothendieck, Le groupe de Brauer, Dix exposés sur la cohomologie des schémas_, Masson and North-Holland, Paris and Amsterdam, (1968), pp. 46–66.
John Duskin, The Azumaya complex of a commutative ring, in: Categorical algebra and its applications (Louvain-La-Neuve, 1987), 107–117, Lecture Notes in Math. 1348, Springer 1988.
Ross Street, Descent, Oberwolfach preprint (sec. 6, Brauer groups) pdf; Some combinatorial aspects of descent theory, Applied categorical structures 12 (2004) 537-576, math.CT/0303175 (sec. 12, Brauer groups)
The relation to cohomology/etale cohomology is discussed in
Ofer Gabber, 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.
Aise Johan de Jong, A result of Gabber (pdf)
Brauer groups of superalgebras are discussed in
C. T. C. Wall, Graded Brauer groups, J. Reine Angew. Math. 213 (1963/1964), 187-199.
Pierre Deligne, Notes on spinors in Quantum Fields and Strings
Peter Donovan, Max Karoubi, Graded Brauer groups and K-theory with local coefficients, Publications Math. IHES 38 (1970), 5-25 (pdf)
Refinement to stable homotopy theory and Brauer ∞-groups is discussed in
Markus Szymik, Brauer spaces for commutative rings and structured ring spectra (arXiv:1110.2956)
Andrew Baker, Birgit Richter, Markus Szymik, Brauer groups for commutative $\mathbb{S}$-algebras, J. Pure Appl. Algebra 216 (2012) 2361–2376 (arXiv:1005.5370)
Related MO discussion includes