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

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

doi:10.1016/j.aim.2019.02.029

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

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

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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 language	English (US)
Pages (from-to)	442-472
Number of pages	31
Journal	Advances in Mathematics
Volume	347
DOIs	https://doi.org/10.1016/j.aim.2019.02.029
State	Published - Apr 30 2019
Externally published	Yes

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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title = "Lech's inequality, the St{\"u}ckrad–Vogel conjecture, and uniform behavior of Koszul homology",

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 St{\"u}ckrad–Vogel in the affirmative. As an application, we obtain results on uniform behavior of the lengths of Koszul homology modules.",

keywords = "Hilbert-Samuel multiplicities, Koszul homology, Lech's inequality, St{\"u}ckrad–Vogel conjecture",

author = "Patricia Klein and Linquan Ma and Quy, {Pham Hung} and Ilya Smirnov and Yongwei Yao",

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: {\textcopyright} 2019",

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AU - Quy, Pham Hung

AU - Smirnov, Ilya

AU - Yao, Yongwei

N1 - 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

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N2 - 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.

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ER -

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

Abstract

Bibliographical note

Keywords

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