Polynomial identity rings as rings of functions

Z. Reichstein; N. Vonessen

doi:10.1016/j.jalgebra.2005.08.008

Polynomial identity rings as rings of functions

Z. Reichstein
, N. Vonessen

Mathematical Sciences

University of British Columbia

Research output: Contribution to journal › Article › peer-review

9 Scopus citations

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.

Original language	English
Pages (from-to)	624-647
Number of pages	24
Journal	Journal of Algebra
Volume	310
Issue number	2
DOIs	https://doi.org/10.1016/j.jalgebra.2005.08.008
State	Published - Apr 15 2007

Funding

* Corresponding author. E-mail addresses: [email protected] (Z. Reichstein), [email protected] (N. Vonessen). URLs: http://www.math.ubc.ca/~reichst (Z. Reichstein), http://www.math.umt.edu/vonessen (N. Vonessen). 1 Author was supported in part by an NSERC research grant. 2 Author gratefully acknowledges the support of the University of Montana and the hospitality of the University of British Columbia during his sabbatical in 2002/2003, when part of this research was done.

Funders
University of British Columbia

Keywords

Central simple algebra
Coordinate ring
Nullstellensatz
Polynomial identity ring
Trace ring

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10.1016/j.jalgebra.2005.08.008

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title = "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 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.",

keywords = "Central simple algebra, Coordinate ring, Nullstellensatz, Polynomial identity ring, Trace ring",

author = "Z. Reichstein and N. Vonessen",

note = "Funding Information: * Corresponding author. E-mail addresses: [email protected] (Z. Reichstein), [email protected] (N. Vonessen). URLs: http://www.math.ubc.ca/\textasciitilde{}reichst (Z. Reichstein), http://www.math.umt.edu/vonessen (N. Vonessen). 1 Author was supported in part by an NSERC research grant. 2 Author gratefully acknowledges the support of the University of Montana and the hospitality of the University of British Columbia during his sabbatical in 2002/2003, when part of this research was done.",

year = "2007",

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doi = "10.1016/j.jalgebra.2005.08.008",

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T1 - Polynomial identity rings as rings of functions

AU - Reichstein, Z.

AU - Vonessen, N.

N1 - Funding Information: * Corresponding author. E-mail addresses: [email protected] (Z. Reichstein), [email protected] (N. Vonessen). URLs: http://www.math.ubc.ca/~reichst (Z. Reichstein), http://www.math.umt.edu/vonessen (N. Vonessen). 1 Author was supported in part by an NSERC research grant. 2 Author gratefully acknowledges the support of the University of Montana and the hospitality of the University of British Columbia during his sabbatical in 2002/2003, when part of this research was done.

PY - 2007/4/15

Y1 - 2007/4/15

N2 - 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.

AB - 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.

KW - Central simple algebra

KW - Coordinate ring

KW - Nullstellensatz

KW - Polynomial identity ring

KW - Trace ring

UR - https://www.scopus.com/pages/publications/33847684848

U2 - 10.1016/j.jalgebra.2005.08.008

DO - 10.1016/j.jalgebra.2005.08.008

M3 - Article

AN - SCOPUS:33847684848

SN - 0021-8693

VL - 310

SP - 624

EP - 647

JO - Journal of Algebra

JF - Journal of Algebra

IS - 2

ER -

Polynomial identity rings as rings of functions

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Keywords

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