Brauer group




For RR a ring, the Brauer group Br(R)Br(R) is the group of Morita equivalence classes of Azumaya algebras over RR.


Relation to categories of modules


For RR a commutative ring, let Alg RAlg_R or 2Vect R2Vect_R (see at 2-vector space/2-module) be the 2-category whose


This may be understood as the 2-category of (generalized) 2-vector bundles over SpecRSpec R, the formally dual space whose function algebra is RR. This is a braided monoidal 2-category.



Br(R)Core(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


This may be understood as the 2-groupoid of (generalized) line 2-bundles over SpecRSpec R (for instance holomorphic line 2-bundles in the case of higher complex analytic geometry), inside that of all 2-vector bundles.


The homotopy groups of Br(R)\mathbf{Br}(R) are the following:

  • π 0(Br(R))\pi_0(\mathbf{Br}(R)) is the Brauer group of RR;

  • π 1(Br(R))\pi_1(\mathbf{Br}(R)) is the Picard group of RR;

  • π 2(Br(R))\pi_2(\mathbf{Br}(R)) is the group of units of RR.

See for instance (Street).


Analogous statements hold for (non-commutative) superalgebras, hence for 2\mathbb{Z}_2-graded algebras. See at superalgebra – Picard 3-group, Brauer group.

Relation to étale cohomology

The Brauer group of a ring RR is a torsion subgroup of the second etale cohomology group of SpecRSpec R with values in the multiplicative group 𝔾 m\mathbb{G}_m

Br(X)H et 2(X,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 et 2(X,𝔾 m)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 et 0(R,𝔾 m)=R ×H^0_{et}(R, \mathbb{G}_m) = R^\times (group of units)

  • H et 1(R,𝔾 m)=Pic(R)H^1_{et}(R, \mathbb{G}_m) = Pic(R) (Picard group: iso classes of invertible RR-modules)

  • H et 2(R,𝔾 m) tor=Br(R)H^2_{et}(R, \mathbb{G}_m)_{tor} = Br(R) (Brauer group:

Morita equivalence classes of Azumaya algebras over RR) (the torsion equivalence classes of the Brauer stack)

It is therefore natural to regard all of H et 2(R,𝔾 m)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 RR a (connective) E-infinity ring (the following is due to Antieau-Gepner 12, see Haugseng 14 for more).

Let GL 1(R)GL_1(R) be its infinity-group of units. If RR is connective, then the first Postnikov stage of the Picard infinity-groupoid

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


B etGL 1() Pic() , \array{ \mathbf{B}_{et} GL_1(-) &\to& Pic(-) \\ && \downarrow \\ && \mathbb{Z} } \,,

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

So H et 1(R,GL 1)H^1_{et}(R, GL_1) is not equal to π 0Pic(R)\pi_0 Pic(R), but it is off only by H et 0(R,)= componentsofRH^0_{et}(R, \mathbb{Z}) = \prod_{components of R} \mathbb{Z}.

Let Mod RMod_R be the (infinity,1)-category of RR-modules.

There is a notion of Mod RMod_R-enriched (infinity,1)-category, of “RR-linear (,1)(\infty,1)-categories”.

Cat RMod RCat_R \coloneqq Mod_R-modules in presentable (infinity,1)-categories.

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

Alg RModCat R Alg_R \stackrel{Mod}{\to} Cat_R

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

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

One shows (Antieau-Gepner 12) that this is exactly the Azumaya RR-algebras modulo Morita equivalence.

Theorem (B. Antieau, D. Gepner)

  1. For RR a connective E E_\infty ring, any Azumaya RR-algebra AA is étale locally trivial: there is an etale cover RSR \to S such that A RSMoritaSA \wedge_R S \stackrel{Morita \simeq}{\to} S.

    (Think of this as saying that an Azumaya RR-algebra is étale-locally a Matrix algebra, hence Morita-trivial: a “bundle of compact operators” presenting a (torsion) GL 1(R)GL_1(R)-2-bundle).

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


  1. BrBr is connected. Hence BrB etΩBrBr \simeq \mathbf{B}_{et} \Omega Br .

  2. ΩBrPic\Omega Br \simeq Pic, hence BrB etPicBr \simeq \mathbf{B}_{et} Pic

Postnikov tower for GL 1(R)GL_1(R):

forn>0:π nGL 1(S)π n for\; n \gt 0: \pi_n GL_1(S) \simeq \pi_n

hence for RSR \to S étale

π nSπ nR π 0Rπ 0S \pi_n S \simeq \pi_n R \otimes_{\pi_0 R} \pi_0 S

This is a quasi-coherent sheaf on π 0R\pi_0 R of the form N˜\tilde N (quasicoherent sheaf associated with a module), for NN an π 0R\pi_0 R-module. By vanishing theorem of higher cohomology for quasicoherent sheaves

H et 1(π 0R,N˜)=0;forp>0 H_{et}^1(\pi_0 R, \tilde N) = 0; for p \gt 0

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

H et p(π 0R;π˜ qG)π qpG(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 qthqth Postnikov stage). From this one gets the following.

  • π˜ 0Br*\tilde \pi_0 Br \simeq *

  • π˜ 1Br\tilde \pi_1 Br \simeq \mathbb{Z};

  • π˜ 2Brπ˜ 1Picπ 0GL 1𝔾 m\tilde \pi_2 Br \simeq \tilde \pi_1 Pic \simeq \pi_0 GL_1 \simeq \mathbb{G}_m

  • π˜ nBr\tilde \pi_n Br is quasicoherent for n>2n \gt 2.

there is an exact sequence

0H et 2(π 0R,𝔾 m)π 0Br(R)H et 1(π 0R,)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 Br(π 0R)H et 2(π 0R,𝔾 m)Br(\pi_0 R) \hookrightarrow H_{et}^2(\pi_0 R, \mathbb{G}_m))

this is split exact and so computes π 0Br(R)\pi_0 Br(R) for connective RR.

Now some more on the case that RR is not connective.

Suppose there exists RϕSR \stackrel{\phi}{\to} S which is a faithful Galois extension for GG a finite group.


  1. (real into complex K-theory spectrum) KOKUKO \to KU (this is 2\mathbb{Z}_2)

  2. tmftmf(3)\to tmf(3)

Give RSR \to S, have a fiber sequence

Gl 1(R/S)fibGL 1(R)GL 1(S)Pic(R/S)fibPic(R)Pic(S)Br(R/S)fibBr(R)Br(S) 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 GG-Galois extension RS hGR \stackrel{\simeq}{\to} S^{hG} (homotopy fixed points)

  1. Mod RMod S hGMod_R \stackrel{\simeq}{\to} Mod_S^{hG}

  2. Alg RAlg S hGAlg_R \stackrel{\simeq}{\to} Alg_S^{hG}

it follows that there is a homotopy fixed points spectral sequence

H p(G,π Σ nGL 1(S))π nGL 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 KOKO-algebra QQ which is a nontrivial element in Br(KO)Br(KO) but becomes trivial in Br(KU)Br(KU).

moduli spaces of line n-bundles with connection on nn-dimensional XX

nnCalabi-Yau 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=0n = 0unit in structure sheafmultiplicative group/group of unitsformal multiplicative groupcomplex K-theory
n=1n = 1elliptic curveline bundlePicard group/Picard schemeJacobianformal Picard groupelliptic cohomology3d Chern-Simons theory/WZW model
n=2n = 2K3 surfaceline 2-bundleBrauer groupintermediate Jacobianformal Brauer groupK3 cohomology
n=3n = 3Calabi-Yau 3-foldline 3-bundleintermediate JacobianCY3 cohomology7d Chern-Simons theory/M5-brane
nnintermediate Jacobian


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. 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 (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

The observation that passing to derived algebraic geometry makes also the non-torsion elements in H et 2(,𝔾 m)H^2_{et}(-,\mathbb{G}_m) be represented by (derived) Azumaya algebras is due to

Related MO discussion includes

Systematic discussion of Brauer groups in derived algebraic geometry is in

Last revised on June 6, 2018 at 07:58:01. See the history of this page for a list of all contributions to it.