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

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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<d}<∞ for d=dim⁡(M)=dim⁡(R).

Original languageEnglish (US)
Pages (from-to)198-224
Number of pages27
JournalJournal of Algebra
Volume543
DOIs
StatePublished - 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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