On certain noncommutative geometries via categories of sheaves of PI-algebras
Lucio Centrone, Maurício Corrêa
TL;DR
The work develops a category based approach to noncommutative geometry by using categories of sheaves of PI algebras to model geometric spaces. It fixes a char 0 field F, a G-graded PI algebra A and a space X, and realizes geometries as objects in the G-var(A) category organized into locally G-graded ringed spaces on X. It develops a presheaf to sheaf formalism, defines stalks and tangent calculi via noncommutative differentials, and extends Morita theory to compare such geometries. The framework covers a wide array of examples including supergeometry, Azumaya algebras, Clifford algebras, and quantum groups at roots of unity, and provides tools for studying Morita invariants and differential calculi in noncommutative settings.
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, which include, for example, the following algebras: 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 sheaf structure belongs to the $G$-graded variety $\text{G-var}(A)$ of algebras generated by $A$, which classify all geometric spaces that belong to $\text{G-var}(A)$. We study conditions to compare two geometries in a (graded) Morita context, as well as their corresponding differential calculi.