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Azumaya algebra

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Definition

Traditional

Given a commutative unital ring RR, an Azumaya RR-algebra is a (noncommutative in general) RR-algebra AA which is finitely generated faithful projective as an RR-module and the canonical morphism A RA opEnd R(A)A\otimes_R A^{op}\to End_R(A) is an isomorphism. This definition extends the notion of a central simple algebra? over a field.

More generally, Grothendieck defines an Azumaya algebra over a scheme XX as a sheaf 𝒜\mathcal{A} of 𝒪 X\mathcal{O}_X-algebras such that for each point xXx\in X, the corresponding stalk 𝒜 x\mathcal{A}_x is an Azumaya 𝒪 X,x\mathcal{O}_{X,x}-algebra.

The Brauer group Br(X)Br(X) classifies Azumaya algebras over XX up to a suitably defined equivalence relation: 𝒜\mathcal{A}\sim\mathcal{B} if 𝒜 𝒪 XEnd()𝒜 𝒪 XEnd()\mathcal{A}\otimes_{\mathcal{O}_X} \mathbf{End}(\mathcal{E}) \cong \mathcal{A}\otimes_{\mathcal{O}_X}\mathbf{End}(\mathcal{F}) for some locally free sheaves of 𝒪 X\mathcal{O}_X-modules \mathcal{E} and \mathcal{F} of finite rank. The group operation of Br(X)Br(X) is induced by the tensor product. The Brauer group can be reexpressed in terms of second nonabelian cohomology; indeed a sheaf of Azumaya algebras over XX determines an 𝒪 X *\mathcal{O}_X^*-gerbe (or U(1)U(1)-gerbe in the manifold context).

Brauer groups and Azumaya algebras are closely related to Morita theory? and they make sense in the context of algebras and bimodules in the context of braided monoidal categories. Karoubi K-theory involves an element in a Brauer group and in the original Karoubi–Donovan paper is related to a twisting with a “local system” which involves Azumaya algebras.

In terms of (derived) étale cohomology

For RR a ring and H et n(,)H^n_{et}(-,-) the etale cohomology, 𝔾 m\mathbb{G}_m the multiplicative group of the affine line; then

  • 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 classes of Azumaya RR-algebras)

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 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.

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

  • G. Cortiñas, Charles Weibel, Homology of Azumaya algebras, Proc. AMS 121, 1, pp. 1994 (jstor)

  • John Duskin, The Azumaya complex of a commutative ring, in Categorical Algebra and its Appl., Lec. Notes in Math. 1348 (1988) doi:10.1007/BFb0081352

  • Alexander Grothendieck, Le groupe de Brauer I, II, III, in Dix exposes sur la cohomologie des schemas (I: Algèbres d’Azumaya et interprétations diverses) North-Holland Pub. Co., Amsterdam (1969)

  • Max Karoubi, Peter Donovan, Graded Brauer groups and KK-theory with local coefficients (pdf)

  • M-A. Knus, M. Ojanguren, Théorie de la descente et algèbres d’Azumaya, Lec. Notes in Math. 389, Springer 1974, doi:10.1007/BFb0057799, MR0417149

  • J. Milne, Étale cohomology, Princeton Univ. Press

  • Ross Street, Descent, Oberwolfach preprint (sec. 6, Brower groups) pdf; Some combinatorial aspects of descent theory, Applied categorical structures 12 (2004) 537-576, math.CT/0303175 (sec. 12, Brower groups)

  • Enrico Vitale, A Picard-Brauer exact sequence of categorical groups, pdf

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

See also

Revised on June 13, 2014 02:59:50 by Urs Schreiber (145.116.130.115)