nLab Azumaya algebra




An Azumaya algebra over a commutative unital ring RR is an algebra over RR that has an inverse up to Morita equivalence. That is, AA is an Azumaya algebra if there is an RR-algebra BB such that B RAB \otimes_R A is Morita equivalent to RR, which is the unit for the tensor product of RR-algebras. Thus, Morita equivalence classes of Azumaya algebras over RR form a group, which is called the Brauer group of RR.



In what follows, RR is a commutative unital ring and algebras over RR are assumed associative and unital but not necessarily commmutative. An Azumaya algebra over RR is an algebra AA over RR obeying any of the following equivalent conditions:

When RR is a field, an Azumaya algebra is the same as a central simple algebra over RR.

For any commutative ring RR there is a monoidal bicategory with

  • algebras over RR as objects,
  • bimodules as morphisms,
  • bimodule homomorphisms as 2-morphisms.

Given any monoidal bicategory we can take its core: that is, the sub-monoidal bicategory where we only keep objects invertible up to equivalence, morphisms invertible up to 2-isomorphism, and invertible 2-morphisms. This core is a 3-group, sometimes called the Picard 3-group, and it has Azumaya algebras over RR as its objects.

Over a scheme

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() 𝒪 XEnd()\mathcal{A}\otimes_{\mathcal{O}_X} \mathbf{End}(\mathcal{E}) \cong \mathcal{B}\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).

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


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.


  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).

Azumaya categories

Borceux and Vitale have noted that the monoidal bicategory of RR-algebras, bimodules and bimodule morphisms can be generalized in the context of enriched category theory, leading to a concept of “Azumaya category”. An Azumaya algebra over the commutative ring RR is then a one-object Azumaya category enriched over RModR Mod.

More precisely, they consider an arbitrary Benabou cosmos VV, meaning a complete and cocomplete symmetric monoidal closed category. This gives a monoidal bicategory VModV Mod with

  • VV-enriched categories as objects,
  • VV-enriched profunctors as morphisms, and
  • VV-natural transformations between VV-enriched profunctors as 2-morphisms.

The core of this monoidal bicategory is a 3-group, and they call the objects of the core Azumaya categories.


  • 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

  • Ana-L. Agore, Stefan Caenepeel?, Gigel Militaru, Braidings on the category of bimodules, Azumaya algebras and epimorphisms of rings, Appl. Categor. Struct. 22, 29–42 (2014) doi

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

The comparison of the Artin’s theorem on characterization of Azumaya algebras and Tomiyama-Takesaki’s theorem on nn-homogeneous C*-algebras is in chapter 9 of

  • Edward Formanek, Noncommutative invariant theory, in: Group actions on rings (Brunswick, Maine, 1984), 87–119, Contemp. Math. 43, Amer. Math. Soc. 1985 doi

See also

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