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

Thomas Hüttemann, Zuhong Zhang

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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.
Original languageEnglish
Pages (from-to)4446-4457
Number of pages12
JournalJournal of Pure and Applied Algebra
Volume223
Issue number10
Early online date28 Jan 2019
DOIs
Publication statusPublished - Oct 2019

Keywords

  • jum loci
  • McCoy rank
  • Betti numbers

ASJC Scopus subject areas

  • Mathematics(all)
  • Algebra and Number Theory

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