Lech's inequality, the Stückrad–Vogel conjecture, and uniform behavior of Koszul homology

Patricia Klein, Linquan Ma, Pham Hung Quy, Ilya Smirnov, Yongwei Yao

Research output: Contribution to journalArticle

1 Scopus citations

Abstract

Let (R,m) be a Noetherian local ring, and let M be a finitely generated R-module of dimension d. We prove that the set [Formula presented] is bounded below by 1/d!e(R‾) where R‾=R/Ann(M). Moreover, when Mˆ is equidimensional, this set is bounded above by a finite constant depending only on M. The lower bound extends a classical inequality of Lech, and the upper bound answers a question of Stückrad–Vogel in the affirmative. As an application, we obtain results on uniform behavior of the lengths of Koszul homology modules.

Original languageEnglish (US)
Pages (from-to)442-472
Number of pages31
JournalAdvances in Mathematics
Volume347
DOIs
StatePublished - Apr 30 2019
Externally publishedYes

Keywords

  • Hilbert-Samuel multiplicities
  • Koszul homology
  • Lech's inequality
  • Stückrad–Vogel conjecture

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