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 $Spec R$ (for instance holomorphic line 2-bundles in the case of higher complex analytic geometry), 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) (see also Grothendieck 64, prop. 1.4 and see at algebraic line n-bundle – Properties). Review discussion is in (Milne, chapter IV). 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 $R$) (the torsion equivalence classes of the Brauer stack)
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” (Taylor 82, Caenepeel-Grandjean 98, Heinloth-Schöer 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.
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 $Br(KU)$.
moduli spaces of line n-bundles with connection on $n$-dimensional $X$
Brauer groups are named after Richard Brauer.
Original discussion includes
Alexander Grothendieck, Le groupe de Brauer : II. Théories cohomologiques. Séminaire Bourbaki, 9 (1964-1966), Exp. No. 297, 21 p. (Numdam)
Alexandre Grothendieck, Le groupe de Brauer, Dix exposés sur la cohomologie des schémas_, Masson and North-Holland, Paris and Amsterdam, (1968), pp. 46–66.
An introduction is in
See also
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)
The “bigger Brauer group” is discussed in
J. Taylor, A bigger Brauer group Pacic J. Math. 103 (1982), 163-203
S. Caenepeel, F. Grandjean, A note on Taylor’s Brauer group. Pacific J. Math. 186 (1998), 13-27
Jochen Heinloth, Stefan Schröer, The bigger Brauer group and twisted sheaves (arXiv:0803.3563)
See also
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
Related MO discussion includes