2004/07/31 by Zinovy Reichstein, Nikolaus Vonessen · 1 citation
Mathematics · #Advanced Topics in Algebra #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #math.AG #math.RA #msc:14A10 #msc:14L30 #msc:16R20 #msc:16R30
paper · pdf · doi:10.1016/j.jalgebra.2005.08.008
published as Journal of Algebra 310 (2007), no. 2, 624--647. · 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
arxiv created 2005/08/05 · openalex publication_date 2006/12/20 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/02
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 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 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.