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Brauer group

Contents

Idea

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

Properties

Relation to categories of modules

Definition

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

Remark

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.

Definition

Let

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

Remark

This may be understood as the 2-groupoid of (generalized) line 2-bundles over SpecRSpec R, inside that of all 2-vector bundles.

Proposition

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).

Example

Analgous 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), 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 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

Relation to derived étale cohomology

More generally, this works for RR a (connective) E-infinity ring (the following is due to Benjamin Antieau and David Gepner).

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

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

is

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-modiles 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 RCar RCat'_R \hookrightarrow Car_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) 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 Matric 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.

Corollary

  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 eétale

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.

Examples

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

  2. tmf tmf(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).

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

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

Related MO discussion includes

Revised on April 1, 2014 10:36:28 by Urs Schreiber (89.204.155.26)