Polynomial identity rings as rings of functions
Abstract
We generalize 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 particular, a prime polynomial identity algebra of degree n is finitely generated if and only if it arises as the "coordinate ring" of a "variety" in this setting. For n = 1 our definitions and results reduce to those of classical affine algebraic geometry.
Cite
@article{arxiv.math/0407152,
title = {Polynomial identity rings as rings of functions},
author = {Zinovy Reichstein and Nikolaus Vonessen},
journal= {arXiv preprint arXiv:math/0407152},
year = {2009}
}
Comments
24 pages. This is the final version of the article, to appear in J. Algebra. Several proofs have been streamlined, and a new section on Brauer-Severi varieties has been added