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 journalArticlepeer-review

2 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

Bibliographical note

Funding Information:
Acknowledgments: The authors would like to thank Mel Hochster and Craig Huneke for valuable discussions, as well as Le Tuan Hoa for his comments. Partial results of this paper were done while the third author was visiting University of Utah. He would like to thank University of Utah for its hospitality and support. The first author has been partially supported by NSF Grants #1401384 and #0943832. The second author is supported in part by NSF Grant #1836867/1600198 and NSF CAREER Grant DMS #1252860/1501102 when preparing this paper. The third author is partially supported by a fund of Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04-2017.10. Finally, the authors would like to thank the referee for his/her comments, which led to Remark 4.14.

Funding Information:
Acknowledgments : The authors would like to thank Mel Hochster and Craig Huneke for valuable discussions, as well as Le Tuan Hoa for his comments. Partial results of this paper were done while the third author was visiting University of Utah. He would like to thank University of Utah for its hospitality and support. The first author has been partially supported by NSF Grants # 1401384 and # 0943832 . The second author is supported in part by NSF Grant # 1836867/1600198 and NSF CAREER Grant DMS # 1252860/1501102 when preparing this paper. The third author is partially supported by a fund of Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04-2017.10 . Finally, the authors would like to thank the referee for his/her comments, which led to Remark 4.14 .

Publisher Copyright:
© 2019

Keywords

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

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