# nLab Brauer group

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

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

## Properties

### Relation to categories of modules

###### Definition

For $R$ a commutative ring, let ${\mathrm{Alg}}_{R}$ or $2{\mathrm{Vect}}_{R}$ (see at 2-vector space/2-module) be the 2-category whose

###### Remark

This may be understood as the 2-category of (generalized) 2-vector bundles over $\mathrm{Spec}R$, the formally dual space whose function algebra is $R$. This is a braided monoidal 2-category.

###### Definition

Let

$\mathrm{Br}\left(R\right)≔\mathrm{Core}\left({\mathrm{Alg}}_{R}\right)$\mathbf{Br}(R) \coloneqq Core(Alg_R)

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

###### Remark

This may be understood as the 2-groupoid of (generalized) line 2-bundles over $\mathrm{Spec}R$, inside that of all 2-vector bundles.

###### Proposition

The homotopy groups of $\mathrm{Br}\left(R\right)$ are the following:

• ${\pi }_{0}\left(\mathrm{Br}\left(R\right)\right)$ is the Brauer group of $R$;

• ${\pi }_{1}\left(\mathrm{Br}\left(R\right)\right)$ is the Picard group of $R$;

• ${\pi }_{2}\left(\mathrm{Br}\left(R\right)\right)$ is the group of units of $R$.

See for instance (Street).

###### Example

Analgous statements hold for (non-commutative) superalgebras, hence for ${ℤ}_{2}$-graded algebras. See at superalgebra – Picard 3-group, Brauer group.

### 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 ${𝔾}_{m}$

$\mathrm{Br}\left(X\right)↪{H}_{\mathrm{et}}^{2}\left(X,{𝔾}_{m}\right)\phantom{\rule{thinmathspace}{0ex}}.$Br(X) \hookrightarrow H^2_{et}(X, \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 ${H}_{\mathrm{et}}^{2}\left(X,{𝔾}_{m}\right)$ 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

• ${H}_{\mathrm{et}}^{0}\left(R,{𝔾}_{m}\right)={R}^{×}$ (group of units)

• ${H}_{\mathrm{et}}^{1}\left(R,{𝔾}_{m}\right)=\mathrm{Pic}\left(R\right)$ (Picard group: iso classes of invertible $R$-modules)

• ${H}_{\mathrm{et}}^{2}\left(R,{𝔾}_{m}{\right)}_{\mathrm{tor}}=\mathrm{Br}\left(R\right)$ (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 ${\mathrm{GL}}_{1}\left(R\right)$ be its infinity-group of units. If $R$ is connective, then the first Postikov stage of the Picard infinity-groupoid

$\mathrm{Pic}\left(R\right)≔\mathrm{Mod}\left(R{\right)}^{×}$Pic(R) \coloneqq Mod(R)^\times

is

$\begin{array}{ccc}{B}_{\mathrm{et}}{\mathrm{GL}}_{1}\left(-\right)& \to & \mathrm{Pic}\left(-\right)\\ & & ↓\\ & & ℤ\end{array}\phantom{\rule{thinmathspace}{0ex}},$\array{ \mathbf{B}_{et} GL_1(-) &\to& Pic(-) \\ && \downarrow \\ && \mathbb{Z} } \,,

where the top morphism is the inclusion of locally free $R$-modules.

so ${H}_{\mathrm{et}}^{1}\left(R,{\mathrm{GL}}_{1}\right)$ is not equal to ${\pi }_{0}\mathrm{Pic}\left(R\right)$, but it is off only by ${H}_{\mathrm{et}}^{0}\left(R,ℤ\right)={\prod }_{\mathrm{components}\mathrm{of}R}ℤ$.

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 $\left(\infty ,1\right)$-categories”.

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

Forming module $\left(\infty ,1\right)$-categories is then an (infinity,1)-functor

${\mathrm{Alg}}_{R}\stackrel{\mathrm{Mod}}{\to }{\mathrm{Cat}}_{R}$Alg_R \stackrel{Mod}{\to} Cat_R

Write $\mathrm{Cat}{\prime }_{R}↪{\mathrm{Car}}_{R}$ for the image of $\mathrm{Mod}$. Then define the Brauer infinity-group to be

$\mathrm{Br}\left(R\right)≔\left(\mathrm{Cat}{\prime }_{R}{\right)}^{×}$Br(R) \coloneqq (Cat'_R)^\times

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 ${E}_{\infty }$ 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) ${\mathrm{GL}}_{1}\left(R\right)$-2-bundle).

2. $\mathrm{Br}:{\mathrm{CAlg}}_{R}^{\ge 0}\to {\mathrm{Gpd}}_{\infty }$ 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 ${\mathrm{GL}}_{1}\left(R\right)$:

$\mathrm{for}\phantom{\rule{thickmathspace}{0ex}}n>0:{\pi }_{n}{\mathrm{GL}}_{1}\left(S\right)\simeq {\pi }_{n}$for\; n \gt 0: \pi_n GL_1(S) \simeq \pi_n

hence for $R\to S$ eétale

${\pi }_{n}S\simeq {\pi }_{n}R{\otimes }_{{\pi }_{0}R}{\pi }_{0}S$\pi_n S \simeq \pi_n R \otimes_{\pi_0 R} \pi_0 S

This is a quasi-coherent sheaf on ${\pi }_{0}R$ of the form $\stackrel{˜}{N}$ (quasicoherent sheaf associated with a module), for $N$ an ${\pi }_{0}R$-module. By vanishing theorem of higher cohomology for quasicoherent sheaves

${H}_{\mathrm{et}}^{1}\left({\pi }_{0}R,\stackrel{˜}{N}\right)=0;\mathrm{for}p>0$H_{et}^1(\pi_0 R, \tilde N) = 0; for p \gt 0

For every (infinity,1)-sheaf $G$ of infinity-groups, there is a spectral sequence

${H}_{\mathrm{et}}^{p}\left({\pi }_{0}R;{\stackrel{˜}{\pi }}_{q}G\right)⇒{\pi }_{q-p}G\left(R\right)$H_{et}^p(\pi_0 R; \tilde \pi_q G) \Rightarrow \pi_{q-p} G(R)

(the second argument on the left denotes the $\mathrm{qth}$ Postnikov stage). From this one gets the following.

• ${\stackrel{˜}{\pi }}_{0}\mathrm{Br}\simeq *$

• ${\stackrel{˜}{\pi }}_{1}\mathrm{Br}\simeq ℤ$;

• ${\stackrel{˜}{\pi }}_{2}\mathrm{Br}\simeq {\stackrel{˜}{\pi }}_{1}\mathrm{Pic}\simeq {\pi }_{0}{\mathrm{GL}}_{1}\simeq {𝔾}_{m}$

• ${\stackrel{˜}{\pi }}_{n}\mathrm{Br}$ is quasicoherent for $n>2$.

there is an exact sequence

$0\to {H}_{\mathrm{et}}^{2}\left({\pi }_{0}R,{𝔾}_{m}\right)\to {\pi }_{0}\mathrm{Br}\left(R\right)\to {H}_{\mathrm{et}}^{1}\left({\pi }_{0}R,ℤ\right)\to 0$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

(notice the inclusion $\mathrm{Br}\left({\pi }_{0}R\right)↪{H}_{\mathrm{et}}^{2}\left({\pi }_{0}R,{𝔾}_{m}\right)$)

this is split exact and so computes ${\pi }_{0}\mathrm{Br}\left(R\right)$ 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 ${ℤ}_{2}$)

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

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

${\mathrm{Gl}}_{1}\left(R/S\right)\stackrel{\mathrm{fib}}{\to }{\mathrm{GL}}_{1}\left(R\right)\to {\mathrm{GL}}_{1}\left(S\right)\to \mathrm{Pic}\left(R/S\right)\stackrel{\mathrm{fib}}{\to }\mathrm{Pic}\left(R\right)\to \mathrm{Pic}\left(S\right)\to \mathrm{Br}\left(R/S\right)\stackrel{\mathrm{fib}}{\to }\mathrm{Br}\left(R\right)\to \mathrm{Br}\left(S\right)\to \cdots$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

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

${H}^{p}\left(G,{\pi }_{•}{\Sigma }^{n}{\mathrm{GL}}_{1}\left(S\right)\right)⇒{\pi }_{-n}{\mathrm{GL}}_{1}\left(S\right)$H^p(G, \pi_\bullet \Sigma^n GL_1(S)) \Rightarrow \pi_{-n} GL_1(S)

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

## 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

Related MO discussion:

Revised on May 7, 2013 23:18:49 by Urs Schreiber (67.216.17.3)