Contents

Idea

For $R$ a ring, the Brauer group $\mathrm{Br}(R)$ is the group of Morita equivalence classes of Azumaya algebras over $R$.

Properties

Relation to categories of modules

See for instance (Street).

Relation to étale cohomology

The Brauer group of a ring $R$ is a torsion subgroup of the second etale cohomology group of $\mathrm{Spec}R$ with values in the multiplicative group ${𝔾}_m$

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 $H_{\mathrm{et}}^2(X,{𝔾}_m)$ 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

• $H_{\mathrm{et}}^0(R,{𝔾}_m)=R^{\times }$ (group of units)

• $H_{\mathrm{et}}^1(R,{𝔾}_m)=\mathrm{Pic}(R)$ (Picard group: iso classes of invertible $R$-modules)

• $H_{\mathrm{et}}^2(R,{𝔾}_m{)}_{\mathrm{tor}}=\mathrm{Br}(R)$ (Brauer group: Morita equivalence classes of Azumaya algebras over $R$)

Relation to derived étale cohomology

More generally, this works for $R$ a (connective) E-infinity ring (the following is due to Benjamin Antieau and David Gepner).

Let ${\mathrm{GL}}_1(R)$ 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 $H_{\mathrm{et}}^1(R,{\mathrm{GL}}_1)$ is not equal to ${\pi }_0\mathrm{Pic}(R)$, but it is off only by $H_{\mathrm{et}}^0(R,\mathbb{Z})={\prod }_{\mathrm{components}\mathrm{of}R}\mathbb{Z}$.

Let ${\mathrm{Mod}}_R$ be the (infinity,1)-category of $R$-modules.

There is a notion of ${\mathrm{Mod}}_R$-enriched (infinity,1)-category, of ”$R$-linear $(\mathrm{\infty },1)$-categories”.

${\mathrm{Cat}}_R≔{\mathrm{Mod}}_R$-modiles in presentable (infinity,1)-categories.

Forming module $(\mathrm{\infty },1)$-categories is then an (infinity,1)-functor

Write $\mathrm{Cat}{\prime }_R\hookrightarrow {\mathrm{Car}}_R$ for the image of $\mathrm{Mod}$. 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)

1. For $R$ a connective $E_{\mathrm{\infty }}$ ring, any Azumaya $R$-algebra $A$ is étale locally trivial: there is an etale cover $R\to S$ such that $A{\wedge }_{R}S\stackrel{\mathrm{Morita}\simeq }{\to }S$.

   (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) ${\mathrm{GL}}_1(R)$-2-bundle).

2. $\mathrm{Br}:{\mathrm{CAlg}}_R^{\ge 0}\to {\mathrm{Gpd}}_{\mathrm{\infty }}$ is a sheaf for the etale cohomology.

Corollary

1. $\mathrm{Br}$ is connected. Hence $\mathrm{Br}\simeq B_{\mathrm{et}}\Omega \mathrm{Br}$.

2. $\Omega \mathrm{Br}\simeq \mathrm{Pic}$, hence $\mathrm{Br}\simeq B_{\mathrm{et}}\mathrm{Pic}$

Postnikov tower for ${\mathrm{GL}}_1(R)$:

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 $\mathrm{qth}$ Postnikov stage). From this one gets the following.

• ${\tilde{\pi }}_0\mathrm{Br}\simeq *$

• ${\tilde{\pi }}_1\mathrm{Br}\simeq \mathbb{Z}$;

• ${\tilde{\pi }}_2\mathrm{Br}\simeq {\tilde{\pi }}_1\mathrm{Pic}\simeq {\pi }_0{\mathrm{GL}}_1\simeq {𝔾}_m$

• ${\tilde{\pi }}_n\mathrm{Br}$ is quasicoherent for $n>2$.

there is an exact sequence

(notice the inclusion $\mathrm{Br}({\pi }_{0}R)\hookrightarrow H_{\mathrm{et}}^2({\pi }_{0}R,{𝔾}_m)$)

this is split exact and so computes ${\pi }_0\mathrm{Br}(R)$ for connective $R$.

Now some more on the case that $R$ is not connective.

Suppose there exists $R\stackrel{\varphi }{\to }S$ which is a faithful Galois extension for $G$ a finite group.

Examples

1. (real into complex K-theory spectrum) $\mathrm{KO}\to \mathrm{KU}$ (this is ${\mathbb{Z}}_2$)

2. tmf $\to \mathrm{tmf}(3)$

Give $R\to S$, have a fiber sequence

Theorem (descent theorems) (Tyler Lawson, David Gepner) Given $G$-Galois extension $R\stackrel{\simeq }{\to }{S}^{\mathrm{hG}}$ (homotopy fixed points?)

1. ${\mathrm{Mod}}_R\stackrel{\simeq }{\to }{\mathrm{Mod}}_S^{\mathrm{hG}}$

2. ${\mathrm{Alg}}_R\stackrel{\simeq }{\to }{\mathrm{Alg}}_S^{\mathrm{hG}}$

it follows that there is a homotopy fixed points spectral sequence

Conjecture The spectral sequence gives an Azumaya $\mathrm{KO}$-algebra $Q$ which is a nontrivial element in $\mathrm{Br}(\mathrm{KO})$ but becomes trivial in $\mathrm{Br}(\mathrm{KU})$.

Related concepts

• group of units/multiplicative group, Picard group

• Azumaya algebra

References

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

• James Milne, Étale cohomology, Princeton Mathematical Series, vol. 33, Princeton University Press, Princeton, New Jersey (1980)

• Jean Giraud, Cohomologie non abelienne, Die Grundlehren der mathematischen Wissenschaften, vol. 179, Springer- Verlag, Berlin, 1971.

• 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)

• Inta Bertuccioni, Brauer groups and cohomology, Archiv der Mathematik, vol. 84 Number 5 (2005)

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)

Related MO discussion:

• Brauer groups and K-theory