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Brauer group

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

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

• objects are $R$-algebras;

• morphisms are bimodules over $R$-algebras;

• 2-morphisms are bimodule homomorphisms.

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

$$\mathrm{Br}(R)≔\mathrm{Core}({\mathrm{Alg}}_R)$$\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

• objects are algebras which are invertible up to Morita equivalence under tensor product;

• morphisms are Morita equivalences;

• 2-morphisms are invertible bimodule homomorphisms.

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 $\mathrm{Br}(R)$ are the following:

• ${\pi }_0(\mathrm{Br}(R))$ is the Brauer group of $R$;

• ${\pi }_1(\mathrm{Br}(R))$ is the Picard group of $R$;

• ${\pi }_2(\mathrm{Br}(R))$ is the group of units of $R$.

See for instance (Street).

Example

Analgous statements hold for (non-commutative) superalgebras, hence for ${\mathbb{Z}}_2$-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 ${𝔾}_m$

$$\mathrm{Br}(X)\hookrightarrow H_{\mathrm{et}}^2(X,{𝔾}_m).$$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 $H_{\mathrm{et}}^2(X,{𝔾}_m)$ 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

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

is

$$\begin{array}{ccc}{B}_{\mathrm{et}}{\mathrm{GL}}_1(-)& \to & \mathrm{Pic}(-)\\ & & \downarrow \\ & & \mathbb{Z}\end{array},$$\array{ \mathbf{B}_{et} GL_1(-) &\to& Pic(-) \\ && \downarrow \\ && \mathbb{Z} } \,,

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

so $H_{\mathrm{et}}^1(R,{\mathrm{GL}}_1)$ 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

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

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

$$\mathrm{Br}(R)≔(\mathrm{Cat}{\prime }_R{)}^{\times }$$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 $E_{\mathrm{\infty }}$ 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 ${\mathrm{GL}}_1(R)$:

$$\mathrm{for}n>0:{\pi }_n{\mathrm{GL}}_1(S)\simeq {\pi }_n$$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 $\tilde{N}$ (quasicoherent sheaf associated with a module), for $N$ an ${\pi }_{0}R$-module. By vanishing theorem of higher cohomology for quasicoherent sheaves

$$H_{\mathrm{et}}^1({\pi }_{0}R,\tilde{N})=0;\mathrm{for}p>0$$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

$$H_{\mathrm{et}}^p({\pi }_{0}R;{\tilde{\pi }}_{q}G)\Rightarrow {\pi }_{q-p}G(R)$$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.

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

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

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

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

there is an exact sequence

$$0\to H_{\mathrm{et}}^2({\pi }_{0}R,{𝔾}_m)\to {\pi }_0\mathrm{Br}(R)\to H_{\mathrm{et}}^1({\pi }_{0}R,\mathbb{Z})\to 0$$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 $\mathrm{Br}({\pi }_{0}R)\hookrightarrow H_{\mathrm{et}}^2({\pi }_{0}R,{𝔾}_m)$)

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

$${\mathrm{Gl}}_1(R/S)\stackrel{\mathrm{fib}}{\to }{\mathrm{GL}}_1(R)\to {\mathrm{GL}}_1(S)\to \mathrm{Pic}(R/S)\stackrel{\mathrm{fib}}{\to }\mathrm{Pic}(R)\to \mathrm{Pic}(S)\to \mathrm{Br}(R/S)\stackrel{\mathrm{fib}}{\to }\mathrm{Br}(R)\to \mathrm{Br}(S)\to \cdots$$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

$$H^p(G,{\pi }_{•}{\Sigma }^n{\mathrm{GL}}_1(S))\Rightarrow {\pi }_{-n}{\mathrm{GL}}_1(S)$$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 $\mathrm{Br}(\mathrm{KO})$ 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