The University of Montana
Department of Mathematical Sciences

Technical report #30/2010

Polynomial Identity Rings as Rings of Functions, II

Nikolaus Vonessen

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 PGLn-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.

Keywords: Polynomial identity ring, central simple algebra, trace ring, coordinate ring, the Nullstellensatz, n-variety.

2010 Mathematics Subject Classification. Primary: 16R30, 16R20; Secondary 14L30, 14A10.

Download Technical Report: Pdf (222 KB) (Revised November 12, 2011)