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

Abstract

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.