Algebraic jump loci for rank and Betti numbers over Laurent polynomial rings

Thomas Hüttemann, Zuhong Zhang

Research output: Contribution to journalArticle

Abstract

Let C be a chain complex of finitely generated free modulesover a commutative Laurent polynomial ring Ls in sindeterminates. Given a group homomorphism ➝p:Zs➝Zt we let p!(C)=C⊗LsLt denote the resulting induced complex over the Laurent polynomial ring Lt in t indeterminates. We prove that the Betti number jump loci, that is, the sets of those homomorphisms p such that bk(p!(C))>bk(C), have a surprisingly simple structure. We allow non-unital commutative rings of coefficients, and work with a notion of Betti numbers that generalises both the usual one for integral domains, and the analogous concept involving McCoy ranks in case of unital commutative rings.
LanguageEnglish
Pages4446-4457
Number of pages12
JournalJournal of Pure and Applied Algebra
Volume223
Issue number10
Early online date28 Jan 2019
DOIs
Publication statusPublished - Oct 2019

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Laurent Polynomials
Betti numbers
Polynomial ring
Commutative Ring
Locus
Jump
Integral domain
Unital
Homomorphisms
Homomorphism
Finitely Generated
Denote
Generalise
Coefficient
Concepts

Keywords

  • jum loci
  • McCoy rank
  • Betti numbers

Cite this

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Algebraic jump loci for rank and Betti numbers over Laurent polynomial rings. / Hüttemann, Thomas; Zhang, Zuhong.

In: Journal of Pure and Applied Algebra, Vol. 223, No. 10, 10.2019, p. 4446-4457.

Research output: Contribution to journalArticle

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