On certain noncommutative geometries via categories of sheaves of PI-algebras

In this work, we propose to study noncommutative geometry using the language of categories of sheaves of algebras with polynomial identities and their properties, introducing new (graded) noncommutative geometries. These include, for example, superalgebras, $\mathbb{Z}_2^n$-graded superalgebras, Azumaya algebras, Clifford and quaternion algebras, the algebra of upper triangular matrices, quantum groups at roots of unity, and also some NC-schemes. More precisely, fix a group $G$, a $G$-graded associative algebra $A$ over a field $F$ of characteristic $0$, and a topological space $X$. We construct a locally $G$-graded ringed space structure on $X$, where the structure sheaf takes values in the $G$-graded variety $\mathbf{G\text{-}var}(A)$ of algebras generated by $A$. This provides a framework that classifies geometric spaces whose local models belong to $\mathbf{G\text{-}var}(A)$. We study conditions under which two such geometries can be compared in a (graded) Morita context, as well as the compatibility of their corresponding differential calculi. As an application, we prove a Morita-equivariant Betti/Riemann–Hilbert theorem: varying the coefficients along the Morita $(2,1)$-groupoid, the fixed-coefficient equivalences are compatible with transport and hence induce a biequivalence of Grothendieck totals.

💡 Research Summary

The paper proposes a unified framework for non‑commutative geometry based on categories of sheaves of polynomial‑identity (PI) algebras equipped with a grading by an arbitrary group G. Starting with a G‑graded associative algebra A over a characteristic‑zero field F, the authors define the G‑graded variety G‑var(A) as the full subcategory of all G‑graded algebras whose graded polynomial identities contain those of A. This variety is a concrete category whose objects are precisely the algebras generated by A under the same PI constraints.

Given a topological space X, a presheaf F: Op(X) → G‑var(A) assigns to each open set U a G‑graded algebra F(U) belonging to the variety. When the usual sheaf axioms (locality and gluing) hold, (X, 𝒪_X) becomes a locally G‑graded ringed space. The construction generalizes classical schemes (commutative algebras), super‑schemes (Z₂‑graded algebras), Z₂ⁿ‑super‑schemes, Azumaya algebras, Clifford and quaternion algebras, upper‑triangular matrix algebras, and quantum groups at roots of unity, all of which satisfy non‑trivial PI’s and thus fit into a single categorical picture.

A central technical contribution is the analysis of when two such geometries can be compared via a graded Morita context. The authors introduce a bimodule P implementing a Morita equivalence between the structure sheaves of two G‑graded ringed spaces. They prove that the associated transport functor L_X(P)(–)=P⊗_B(–) behaves well on stalks: for each point x∈X there is a canonical B′‑linear isomorphism (P⊗_B E)_x ≅ P⊗_B E_x, natural in the sheaf E. This pointwise description bridges the sheaf‑theoretic and algebraic viewpoints and yields a 2‑categorical (2,1)‑groupoid of Morita equivalences.

Using this machinery the authors establish a Morita‑equivariant version of the Betti/Riemann–Hilbert correspondence. Classical Betti and de Rham (or Riemann–Hilbert) equivalences for a fixed coefficient field are known to be compatible with the usual derived functors. The paper shows that when coefficients are varied along the Morita (2,1)‑groupoid, the fixed‑coefficient equivalences transport coherently, producing a biequivalence of the Grothendieck totals of the corresponding sheaf categories. In other words, the derived category of constructible sheaves with coefficients in any Morita‑equivalent algebra is canonically equivalent to the derived category of regular holonomic D‑modules with the same coefficients, and these equivalences fit together into a single higher‑categorical structure.

The work is illustrated with detailed examples: Z₂‑graded Grassmann algebras (supergeometry), Z₂ⁿ‑graded algebras, Azumaya algebras (central simple algebras), Clifford algebras, upper‑triangular matrix algebras, and quantum groups at roots of unity. For each case the authors identify the relevant PI’s, construct the associated free G‑graded algebra, and describe the resulting sheaf‑theoretic geometry.

Overall, the paper achieves a conceptual synthesis: polynomial identities provide the algebraic backbone, the graded variety G‑var(A) supplies a geometric “moduli” of local models, sheaves on a space turn these local models into global non‑commutative spaces, and graded Morita theory together with the Morita‑equivariant Betti/Riemann–Hilbert theorem supplies the tools for comparing and transporting structures across different coefficient algebras. This framework promises applications in non‑commutative differential geometry, representation theory of quantum groups, and the study of supersymmetric field theories where both bosonic and fermionic degrees of freedom coexist.