nLab Azumaya algebra

Idea

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

Definition

Traditional

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

• There exists an $R$-algebra $B$ such that the tensor product of $R$-algebras $B \otimes_R A$ is Morita equivalent to $R$.

• The $R$-algebra $A^{\mathrm{op}} \otimes_R A$ is Morita equivalent to $R$, where $A^{\mathrm{op}}$ is the opposite algebra of $A$.

• The center of $A$ is $R$, and $A$ is a separable algebra.

• As a left $R$-module, $A$ finitely generated, faithful and projective, and the canonical morphism $A\otimes_R A^{op}\to End_R(A)$ is an isomorphism.

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

For any commutative ring $R$ there is a monoidal bicategory with

• algebras over $R$ 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 $R$ as its objects.

Over a scheme

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

The Brauer group $Br(X)$ classifies Azumaya algebras over $X$ up to a suitably defined equivalence relation: $\mathcal{A}\sim\mathcal{B}$ if $\mathcal{A}\otimes_{\mathcal{O}_X} \mathbf{End}(\mathcal{E}) \cong \mathcal{B}\otimes_{\mathcal{O}_X}\mathbf{End}(\mathcal{F})$ for some locally free sheaves of $\mathcal{O}_X$-modules $\mathcal{E}$ and $\mathcal{F}$ of finite rank. The group operation of $Br(X)$ 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 $X$ determines an $\mathcal{O}_X^*$-gerbe (or $U(1)$-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 $R$ a ring and $H^n_{et}(-,-)$ the etale cohomology, $\mathbb{G}_m$ the multiplicative group of the affine line; then

• $H^0_{et}(R, \mathbb{G}_m) = R^\times$ (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 classes of Azumaya $R$-algebras)

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

Let $GL_1(R)$ be its infinity-group of units. If $R$ is connective, then the first Postikov 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”.

$Cat_R \coloneqq Mod_R$-modiles in presentable (infinity,1)-categories.

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

$$Alg_R \stackrel{Mod}{\to} Cat_R$$

Write $Cat'_R \hookrightarrow Car_R$ for the image of $Mod$. Then define the Brauer infinity-group to be

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

One shows (Antieau-Gepner) 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 Matric 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 $Br(KU)$.

Azumaya categories

Borceux and Vitale have noted that the monoidal bicategory of $R$-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 $R$ is then a one-object Azumaya category enriched over $R Mod$.

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

• $V$-enriched categories as objects,
• $V$-enriched profunctors as morphisms, and
• $V$-natural transformations between $V$-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.

• Francis Borceux, Enrico Vitale, Azumaya categories, Applied Categorical Structures 10 (2002), 449-467. (pdf)

Related concepts

• group of units, Picard group, Brauer group, Picard 3-group
• homogeneous C-star-algebra, continuous-trace algebra
• special case: central simple algebra

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 $K$-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 group” $H^2_{et}(-,\mathbb{G}_m)$ be represented by (derived) Azumaya algebras is due to

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

• Michael Artin, On Azumaya algebras and finite-dimensional representations of rings, J. Algebra 11, 532–563 doi

The comparison of the Artin’s theorem on characterization of Azumaya algebras and Tomiyama-Takesaki’s theorem on $n$-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

• Wikipedia, Azumaya algebra

• Urs Schreiber, Picard and Brauer 2-groups, String Theory Coffee Table, 2006.

• John Baez, The Brauer 3-group, $n$-Category Café, 2020.