nLab Brauer group

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

• 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 $Spec R$, 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

• 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 $Spec R$ (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

Analogous 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 $Br(KU)$.

Related concepts

• Brauer stack

• formal Brauer group

• group of units/multiplicative group, Picard group

• Azumaya algebra

• Brauer ∞-group

moduli spaces of line n-bundles with connection on $n$-dimensional $X$

$n$	Calabi-Yau n-fold	line n-bundle	moduli of line n-bundles	moduli of flat/degree-0 n-bundles	Artin-Mazur formal group of deformation moduli of line n-bundles	complex oriented cohomology theory	modular functor/self-dual higher gauge theory of higher dimensional Chern-Simons theory
$n = 0$		unit in structure sheaf	multiplicative group/group of units		formal multiplicative group	complex K-theory
$n = 1$	elliptic curve	line bundle	Picard group/Picard scheme	Jacobian	formal Picard group	elliptic cohomology	3d Chern-Simons theory/WZW model
$n = 2$	K3 surface	line 2-bundle	Brauer group	intermediate Jacobian	formal Brauer group	K3 cohomology
$n = 3$	Calabi-Yau 3-fold	line 3-bundle		intermediate Jacobian		CY3 cohomology	7d 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)

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. 5, 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)

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)

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

• Rune Haugseng, The higher Morita category of $E_n$-algebras (arXiv:1412.8459)

The “bigger Brauer group” is discussed in

• J. Taylor, A bigger Brauer group Pacific J. Math. 103 (1982), 163-203 (projecteuclid)

• 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

• Jochen Heinloth, Marc Levine, Stefan Scröer, Forschungsseminar: Brauer groups and Artin stack, 07 (pdf)

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

• Bertrand Toën, Derived Azumaya algebras and generators for twisted derived categories (arXiv:1002.2599)

Related MO discussion includes

• Brauer groups and K-theory

Systematic discussion of Brauer groups in derived algebraic geometry is in

• Benjamin Antieau, David Gepner, Brauer groups and étale cohomology in derived algebraic geometry, Geom. Topol. 18 (2014) 1149-1244 (arXiv:1210.0290)

For the Brauer-Picard 2-group of a tensor category, see

• Alexei Davydov, Dmitri Nikshych, Braided Picard groups and graded extensions of braided tensor categories, (arXiv:2006.08022)