# nLab Brauer group

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

For $R$ a ring, the Brauer group $Br(R)$ 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 $Alg_R$ or $2Vect_R$ (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 $Spec R$, the formally dual space whose function algebra is $R$. This is a braided monoidal 2-category.

###### Definition

Let

$\mathbf{Br}(R) \coloneqq Core(Alg_R)$

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 $Spec R$ (for instance holomorphic line 2-bundles in the case of higher complex analytic geometry), inside that of all 2-vector bundles.

###### Proposition

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).

###### 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 $Spec R$ with values in the multiplicative group $\mathbb{G}_m$

$Br(X) \hookrightarrow H^2_{et}(X, \mathbb{G}_m) \,.$

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.

### Relation to derived étale cohomology

More generally, this works for $R$ a (connective) E-infinity ring (the following is due to Antieau-Gepner 12, see Haugseng 14 for more).

Let $GL_1(R)$ be its infinity-group of units. If $R$ is connective, then the first Postnikov stage of the Picard infinity-groupoid

$Pic(R) \coloneqq Mod(R)^\times$

is

$\array{ \mathbf{B}_{et} GL_1(-) &\to& Pic(-) \\ && \downarrow \\ && \mathbb{Z} } \,,$

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

So $H^1_{et}(R, GL_1)$ 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”.

$Cat_R \coloneqq Mod_R$-modules in presentable (infinity,1)-categories.

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

$Alg_R \stackrel{Mod}{\to} Cat_R$

Write $Cat'_R \hookrightarrow Cat_R$ for the image of $Mod$. Then define the Brauer infinity-group to be

$Br(R) \coloneqq (Cat'_R)^\times$

One shows (Antieau-Gepner 12) 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 $A \wedge_R S \stackrel{Morita \simeq}{\to} S$.

(Think of this as saying that an Azumaya $R$-algebra is étale-locally a Matrix algebra, hence Morita-trivial: a “bundle of compact operators” presenting a (torsion) $GL_1(R)$-2-bundle).

2. $Br : CAlg_R^{\geq 0} \to Gpd_\infty$ is a sheaf for the etale cohomology.

Corollary

1. $Br$ is connected. Hence $Br \simeq \mathbf{B}_{et} \Omega Br$.

2. $\Omega Br \simeq Pic$, hence $Br \simeq \mathbf{B}_{et} Pic$

Postnikov tower for $GL_1(R)$:

$for\; n \gt 0: \pi_n GL_1(S) \simeq \pi_n$

hence for $R \to S$ étale

$\pi_n S \simeq \pi_n R \otimes_{\pi_0 R} \pi_0 S$

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

$H_{et}^1(\pi_0 R, \tilde N) = 0; for p \gt 0$

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

$H_{et}^p(\pi_0 R; \tilde \pi_q G) \Rightarrow \pi_{q-p} G(R)$

(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

$0 \to H_{et}^2(\pi_0 R, \mathbb{G}_m) \to \pi_0 Br(R) \to H_{et}^1(\pi_0 R, \mathbb{Z}) \to 0$

(notice the inclusion $Br(\pi_0 R) \hookrightarrow H_{et}^2(\pi_0 R, \mathbb{G}_m)$)

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

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

2. tmf $\to tmf(3)$

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

$Gl_1(R/S) \stackrel{fib}{\to} GL_1(R) \to GL_1(S) \to Pic(R/S) \stackrel{fib}{\to} Pic(R) \to Pic(S) \to Br(R/S) \stackrel{fib}{\to} Br(R) \to Br(S) \to \cdots$

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

1. $Mod_R \stackrel{\simeq}{\to} Mod_S^{hG}$

2. $Alg_R \stackrel{\simeq}{\to} Alg_S^{hG}$

it follows that there is a homotopy fixed points spectral sequence

$H^p(G, \pi_\bullet \Sigma^n GL_1(S)) \Rightarrow \pi_{-n} GL_1(S)$

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 $X$

$n$Calabi-Cau n-foldline n-bundlemoduli of line n-bundlesmoduli of flat/degree-0 n-bundlesArtin-Mazur formal group of deformation moduli of line n-bundlescomplex oriented cohomology theorymodular functor/self-dual higher gauge theory of higher dimensional Chern-Simons theory
$n = 0$unit in structure sheafmultiplicative group/group of unitsformal multiplicative groupcomplex K-theory
$n = 1$elliptic curveline bundlePicard group/Picard schemeJacobianformal Picard groupelliptic cohomology3d Chern-Simons theory/WZW model
$n = 2$K3 surfaceline 2-bundleBrauer groupintermediate Jacobianformal Brauer groupK3 cohomology
$n = 3$Calabi-Yau 3-foldline 3-bundleintermediate JacobianCY3 cohomology7d Chern-Simons theory/M5-brane
$n$intermediate Jacobian

## References

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

• Pete Clark, On the Brauer group (2003) (pdf)

• 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

Refinement to stable homotopy theory and Brauer ∞-groups is discussed in

Unification of all this in a theory of (infinity,n)-modules is in

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)

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