Some extensions of theorems of Knörrer and Herzog–Popescu

Alex S. Dugas, Graham J. Leuschke

Research output: Contribution to journalArticlepeer-review

Abstract

A construction due to Knörrer shows that if N is a maximal Cohen–Macaulay module over a hypersurface defined by f+y2, then the first syzygy of N/yN decomposes as the direct sum of N and its own first syzygy. This was extended by Herzog–Popescu to hypersurfaces f+yn, replacing N/yN by N/yn−1N. We show, in the same setting as Herzog–Popescu, that the first syzygy of N/ykN is always an extension of N by its first syzygy, and moreover that this extension has useful approximation properties. We give two applications. First, we construct a ring Λ over which every finitely generated module has an eventually 2-periodic projective resolution, prompting us to call it a “non-commutative hypersurface ring”. Second, we give upper bounds on the dimension of the stable module category (a.k.a. the singularity category) of a hypersurface defined by a polynomial of the form x1a1+…+xdad.

Original languageEnglish (US)
Pages (from-to)94-120
Number of pages27
JournalJournal of Algebra
Volume571
DOIs
StatePublished - Apr 1 2021

Keywords

  • Huneke issue
  • Hypersurface rings
  • Maximal Cohen–Macaulay modules
  • Stable module category
  • Syzygies

ASJC Scopus subject areas

  • Algebra and Number Theory

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