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.

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

References in corpus (1)

• Group actions on central simple algebras: a geometric approach

Cited by in corpus (9)

• Geometry of free loci and factorization of noncommutative polynomials
• Group actions on central simple algebras: a geometric approach
• Tame group actions on central simple algebras
• Group actions and invariants in algebras of generic matrices
• Factorization of noncommutative polynomials and Nullstellensätze for the free algebra
• Boundaries, Bundles and Trace Algebras
• T-ideals of Cayley Hamilton algebras
• Polynomial identity rings as rings of functions, II
• Central Simple Algebras with Involution: A Geometric Approach