Azumaya Algebra Over Its Center

• Azumaya algebras over their centers are finitely generated, projective algebras that generalize central simple algebras and exhibit a central, separable structure.
• They are étale-locally isomorphic to matrix algebras, allowing precise characterization of the Azumaya locus through discriminant ideals and smooth center properties.
• Applications span PI algebras, quantum groups, and noncommutative geometry, where graded and skew group constructions enhance both structural analysis and representation theory.

An Azumaya algebra over its center is a fundamental object in noncommutative algebra and algebraic geometry, generalizing the notion of central simple algebras over fields to the relative setting of commutative base rings. The interplay between the structure of the algebra, its center, and the corresponding representation theory is crucial in modern studies of polynomial identity (PI) algebras, quantum algebras at roots of unity, and noncommutative algebraic geometry.

1. Definitions and Characterizations

Let $R$ be a commutative ring and $A$ an associative unital $R$-algebra. Two equivalent definitions specify when $A$ is Azumaya over $R$ (and hence, over its center $Z(A)$):

• (Projective plus Endomorphism Isomorphism): $A$ is finitely generated and projective as an $R$-module, and the canonical $R$-algebra morphism

$$\varphi: A\otimes_R A^{\mathrm{op}}\longrightarrow \operatorname{End}_R(A),\qquad a\otimes b \mapsto (x\mapsto axb)$$

is an isomorphism.

• (Locally Matrix for the Étale Topology): $A$ is étale-locally isomorphic to a matrix algebra: there exists an étale cover $\{R\to S_i\}$ such that each $A\otimes_R S_i \cong M_{n_i}(S_i)$ as $S_i$-algebras.

These two conditions are constructively equivalent via faithfully flat descent and étale-local splitting arguments. For a prime PI algebra $A$ that is finite over its center $Z(A)$, the Azumaya locus is the open subset of $\operatorname{Max}Z(A)$ where $A\otimes_{Z(A)} Z(A)_{\mathfrak{m}}$ is Azumaya over $Z(A)_{\mathfrak{m}}$, i.e., $A/\mathfrak{m}A \cong M_n(k)$, with $n$ the PI-degree of $A$ and $k = Z(A)/\mathfrak{m}$ (Coquand et al., 2023, Brown et al., 2017).

2. Structural Properties and the Center

A central consequence of the Azumaya property is that $A$ is central and separable over $Z(A)$: $Z(A) = R$ in the Azumaya setting, and $A$ behaves like a "bundle" of central simple algebras over $\operatorname{Spec}Z(A)$. The center can be characterized as the equalizer of the natural maps $x\mapsto x\otimes 1$ and $x\mapsto 1\otimes x$ from $A$ to $A\otimes_{Z(A)}A$. Descent theory ensures that, étale-locally, the center of $A$ identifies with the base ring (Coquand et al., 2023).

For graded and crossed product constructions, characterization of the center may require explicit analysis of the grading or group action, as in the case of graded Azumaya algebras and skew group rings (Hazrat et al., 2010, Crawford, 2017).

3. Detection and Characterization of the Azumaya Locus

For prime affine PI algebras $A$ finite over their center $Z(A)$, the Azumaya locus can be precisely characterized using discriminant ideals. Given a trace-like map $\operatorname{tr}:A\to Z(A)$, the top discriminant ideal $D_{n^2}(A/Z(A),\operatorname{tr})$, with $n$ the PI-degree, defines a closed subscheme whose complement is the Azumaya locus: $$\operatorname{V}(D_{n^2}(A/Z(A),\operatorname{tr})) = \operatorname{Max}Z(A)\setminus \mathcal{A}(A)$$ Thus, a maximal ideal $\mathfrak{m}$ lies in the Azumaya locus if and only if $$D_{n^2}(A/Z(A),\operatorname{tr})\nsubseteq \mathfrak{m}$$, which is equivalent to the statement that the determinant of the $n^2\times n^2$ trace matrix is nonzero modulo $\mathfrak{m}$. Under further homological conditions (center regular in codimension $1$, $A$ Cohen–Macaulay over its center, finite global dimension), the Azumaya locus coincides with the smooth locus of $\operatorname{Spec}Z(A)$ (Brown et al., 2017).

Specific examples include: quantized Weyl algebras at roots of unity, quantum Euclidean spaces, generalized skein algebras, and $W$-algebras (Shu et al., 2017, Chirvasitu, 6 Aug 2025, Mukherjee, 2020, Karuo et al., 18 Jan 2025).

Criterion	Azumaya Locus Condition	Reference
PI algebra, discriminant	$$D_{n^2}(A/Z(A),\operatorname{tr})\nsubseteq \mathfrak{m}$$	(Brown et al., 2017)
Geometric, smoothness	$\mathfrak{m}$ is a smooth point of $\operatorname{Spec}Z(A)$	(Shu et al., 2017)
Graded, central simple	Azumaya iff graded central simple, finite over center	(Hazrat et al., 2010)
Skew group ring	$A$ Azumaya over $Z(A)$, $G$ acts freely on $\operatorname{Max}Z(A)$	(Crawford, 2017)

4. Examples and Case Studies

Quantum Kleinian Singularities and Skew Group Rings

A skew group ring $A\#G$ (with $A$ prime noetherian, $G$ finite) is Azumaya over its center if (i) $A$ is Azumaya over its own center, and (ii) the $G$-action is "X-outer" and free on $\operatorname{Max}Z(A)$. In the inner cyclic case (cyclic group acting by inner automorphisms), $A$ is Azumaya over $Z(A)$ if and only if $A\#G$ is Azumaya over its center, and the module ranks are preserved (Crawford, 2017).

Quantum Euclidean Spaces

For $O_q(\mathbb{R}^{2n})$ with $q$ a root of unity, the Azumaya locus is the locus where all central parameters associated to normal elements $w_i$ are invertible; at these points, the fiber algebras are central simple, and the representation theory is controlled by the PI-degree (Mukherjee, 2020).

Quantum Spheres and Skein Algebras

For function algebras on quantum spheres and generalized skein algebras at roots of unity, the Azumaya locus is explicitly described in terms of the center and fiber rank conditions. In particular, for rational deformation parameters on quantum spheres, the Azumaya property coincides with algebraic finite generation over the center, and the Azumaya locus is the complement of the "jump complex" locus in the base branched cover (Chirvasitu, 6 Aug 2025). For skein algebras, after localization at an explicit central element (e.g., boundary loop product), the algebra becomes Azumaya over its localized center, and the Azumaya locus parametrizes generic irreducible representations of maximal dimension (Karuo et al., 18 Jan 2025).

5. Azumaya Geometry, Grothendieck Topologies, and Representation Stacks

The category of Azumaya algebras with varying centers can be equipped with generalized Grothendieck topologies, extending the Zariski/étale topologies from schemes to noncommutative and "Azumaya" geometry. Presheaves of representation functors $$\operatorname{Rep}_R(A) = \operatorname{Hom}_{C\text{-alg}}(R,A)$$ with $A$ Azumaya over $C$ yield sheaves for a broad class of Grothendieck topologies. The associated "Azumaya representation schemes" $\operatorname{rep}_A(R)$ are represented by affine $C$-schemes, generalizing classical representation varieties and encapsulating the torsor structure arising from forms of the matrix algebra and moduli of projective $A$-modules (Hemelaer et al., 2016).

6. Graded Azumaya Algebras and K-theory

A $\Gamma$-graded algebra $A$ over a graded commutative ring $R$ is graded Azumaya if it is graded faithfully projective and the canonical map $$A\otimes_R A^{\mathrm{op}} \to \operatorname{End}_R(A)$$ is an isomorphism of graded $R$-algebras. Graded central simple algebras are automatically graded Azumaya, and results such as isomorphism of graded Quillen $K$-groups (after inverting the rank) pass directly from the ungraded case, preserving Morita invariance and structure theory (Hazrat et al., 2010).

7. Representation Theory and Geometric Consequences

At points of the Azumaya locus, fibers $A/\mathfrak{m}A$ are central simple algebras; thus, modules with central character $\mathfrak{m}$ have maximal possible dimension (equal to the PI-degree), and are classified up to isomorphism by their central character. This underlies the connection between the Azumaya locus and the smooth/regular locus of the center, describing a generic regime for irreducible representations with full matrix algebra structure (Shu et al., 2017, Karuo et al., 18 Jan 2025, Mukherjee, 2020).

A notable phenomenon—exemplified in finite $W$-algebras and quantum Kleinian singularities—is that the Azumaya locus over the center coincides with the smooth locus of the center, enabling a tight correlation between noncommutative and commutative singularity theory (Shu et al., 2017, Brown et al., 2017).

• "Azumaya algebras and Barr Theorem" (Coquand et al., 2023)
• "Azumaya loci and discriminant ideals of PI algebras" (Brown et al., 2017)
• "Centers and Azumaya loci of finite W-algebras" (Shu et al., 2017)
• "Azumaya geometry and representation stacks" (Hemelaer et al., 2016)
• "On Graded Simple Algebras" (Hazrat et al., 2010)
• "Azumaya skew group algebras and an application to quantum Kleinian singularities" (Crawford, 2017)
• "Azumaya Loci Of The Quantum Euclidean $2n$-Space" (Mukherjee, 2020)
• "Polynomial identities and Azumaya loci for rational quantum spheres" (Chirvasitu, 6 Aug 2025)
• "Center of generalized skein algebras" (Karuo et al., 18 Jan 2025)