Polynomial identity rings as rings of functions, II

TL;DR

This paper extends the non-commutative geometric framework relating prime polynomial identity algebras and matrix-valued functions from characteristic zero to prime characteristic, correcting previous errors and characterizing prime PI-algebras.

Contribution

It generalizes the non-commutative geometric correspondence to prime characteristic and characterizes prime PI-algebras as coordinate rings of non-commutative varieties.

Findings

Extension of the framework to prime characteristic.

Correction of a previous mistake in the theory.

Prime PI-algebras of degree n are coordinate rings of n-varieties.

Abstract

In characteristic zero, Zinovy Reichstein and the author generalized the usual relationship between irreducible Zariski closed subsets of the affine space, their defining ideals, coordinate rings, and function fields, to a non-commutative setting, where "varieties" carry a PGL_n-action, regular and rational "functions" on them are matrix-valued, "coordinate rings" are prime polynomial identity algebras, and "function fields" are central simple algebras of degree n. In the present paper, much of this is extended to prime characteristic. In addition, a mistake in the earlier paper is corrected. One of the results is that the finitely generated prime PI-algebras of degree n are precisely the rings that arise as "coordinate rings" of "n-varieties" in this setting. For n = 1 the definitions and results reduce to those of classical affine algebraic geometry.