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Polynomial identity rings as rings of functions

Zinovy Reichstein·Nikolaus Vonessen
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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.

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Zinovy Reichstein, Nikolaus Vonessen. 2005-08-05. Polynomial identity rings as rings of functions. https://doi.org/10.1016/j.jalgebra.2005.08.008

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