Polynomial identity rings as rings of functions, II

Details Polynomial identity rings as rings of functions, II Polynomial identity rings as rings of functions, II

2010-12-04

arxiv.org/abs/1012.0869v2

Mathematics

Rings and Algebras

24 pages, LaTeX. Many changes. Theorem II.1.3 has been strengthened, Sections II.6-II.8 have been rewritten, and Section II.9

Scientific paper

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.

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