Characterizing finite length local cohomology in terms of bounds on Koszul cohomology

Patricia Klein

doi:10.1016/j.jalgebra.2019.09.026

Characterizing finite length local cohomology in terms of bounds on Koszul cohomology

Patricia Klein

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Let (R,m,κ) be a local ring. We give a characterization of R-modules M whose local cohomology is finite length up to some index in terms of asymptotic vanishing of Koszul cohomology on parameter ideals up to the same index. In particular, we show that a quasi-unmixed module M is asymptotically Cohen-Macaulay if and only if M is Cohen-Macaulay on the punctured spectrum if and only if sup⁡{ℓ(Hⁱ(f₁,…,f_d;M))|f₁,…,f_d=m, i<d}<∞ for d=dim⁡(M)=dim⁡(R).

Original language	English (US)
Pages (from-to)	198-224
Number of pages	27
Journal	Journal of Algebra
Volume	543
DOIs	https://doi.org/10.1016/j.jalgebra.2019.09.026
State	Published - Feb 1 2020

Bibliographical note

Funding Information:
The author would like to thank her advisor, Mel Hochster, for his tremendous support during her doctoral work, of which this paper is part. The author has been partially supported by NSF Grants DMS-1401384 and DMS-0943832.

Publisher Copyright:
© 2019 Elsevier Inc.

Keywords

Hilbert-Samuel multiplicities
Koszul cohomology
Lech's inequality
Local cohomology

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title = "Characterizing finite length local cohomology in terms of bounds on Koszul cohomology",

abstract = "Let (R,m,κ) be a local ring. We give a characterization of R-modules M whose local cohomology is finite length up to some index in terms of asymptotic vanishing of Koszul cohomology on parameter ideals up to the same index. In particular, we show that a quasi-unmixed module M is asymptotically Cohen-Macaulay if and only if M is Cohen-Macaulay on the punctured spectrum if and only if sup⁡{ℓ(Hi(f1,…,fd;M))|f1,…,fd=m, i",

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N2 - Let (R,m,κ) be a local ring. We give a characterization of R-modules M whose local cohomology is finite length up to some index in terms of asymptotic vanishing of Koszul cohomology on parameter ideals up to the same index. In particular, we show that a quasi-unmixed module M is asymptotically Cohen-Macaulay if and only if M is Cohen-Macaulay on the punctured spectrum if and only if sup⁡{ℓ(Hi(f1,…,fd;M))|f1,…,fd=m, i

AB - Let (R,m,κ) be a local ring. We give a characterization of R-modules M whose local cohomology is finite length up to some index in terms of asymptotic vanishing of Koszul cohomology on parameter ideals up to the same index. In particular, we show that a quasi-unmixed module M is asymptotically Cohen-Macaulay if and only if M is Cohen-Macaulay on the punctured spectrum if and only if sup⁡{ℓ(Hi(f1,…,fd;M))|f1,…,fd=m, i

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KW - Lech's inequality

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Characterizing finite length local cohomology in terms of bounds on Koszul cohomology

Abstract

Bibliographical note

Keywords

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