k summands of syzygies over rings of positive Burch index via canonical resolutions

Michael DeBellevue, Claudia Miller

Research output: Contribution to journalArticlepeer-review

Abstract

In recent work, Dao and Eisenbud define the notion of a Burch index, expanding the notion of Burch rings of Dao, Kobayashi, and Takahashi, and show that for any module over a ring of Burch index at least 2, its nth syzygy contains direct summands of the residue field for n=4 or 5 and all n≥7. We investigate how this behavior is explained by the bar resolution formed from appropriate differential graded (dg) resolutions, yielding a new proof that includes all n≥5, which is sharp. When the module is Golod, we use instead the bar resolution formed from A resolutions to identify such k summands explicitly for all n≥4 and show that the number of these grows exponentially as the homological degree increases.

Original languageEnglish (US)
Pages (from-to)657-672
Number of pages16
JournalJournal of Algebra
Volume666
DOIs
StatePublished - Mar 15 2025

Keywords

  • Bar resolution
  • Burch index
  • Dg and A-infinity structures
  • Syzygy

ASJC Scopus subject areas

  • Algebra and Number Theory

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