Nonnoetherian geometry

Charlie R Beil

Research output: Contribution to journalArticle (Academic Journal)peer-review

6 Citations (Scopus)
128 Downloads (Pure)

Abstract

We introduce a theory of geometry for nonnoetherian commutative algebras with finite Krull dimension. In particular, we establish new notions of normalization and height: depiction (a special noetherian overring) and geometric codimension. The resulting geometries are algebraic varieties with positive dimensional points, and are thus inherently nonlocal. These notions also give rise to new equivalent characterizations of noetherianity that are primarily geometric.

We then consider an application to quiver algebras whose simple modules of maximal dimension are one dimensional at each vertex. We show that the vertex corner rings of A are all isomorphic if and only if A is noetherian, if and only if the center Z of A is noetherian, if and only if A is a finitely generated Z-module. Furthermore, we show that Z is depicted by a commutative algebra generated by the cycles in its quiver. We conclude with an example of a quiver algebra where projective dimension and geometric codimension, rather than height, coincide.
Original languageEnglish
Article number1650176
Number of pages26
JournalJournal of Algebra and Its Applications
Volume15
Issue number9
Early online date6 Jan 2016
DOIs
Publication statusPublished - Nov 2016

Keywords

  • Nonnoetherian rings
  • foundations of algebraic geometry
  • noncommutative algebraic geometry

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