The cohen-macaulay property in derived commutative algebra

Liran Shaul

Research output: Contribution to journalArticlepeer-review

10 Scopus citations

Abstract

By extending some basic results of Grothendieck and Foxby about local cohomology to commutative DG-rings, we prove new amplitude inequalities about finite DG-modules of finite injective dimension over commutative local DG-rings, complementing results of Jørgensen and resolving a recent conjecture of Minamoto. When these inequalities are equalities, we arrive at the notion of a local-Cohen-Macaulay DG-ring. We make a detailed study of this notion, showing that much of the classical theory of Cohen-Macaulay rings and modules can be generalized to the derived setting, and that there are many natural examples of local-Cohen-Macaulay DG-rings. In particular, local Gorenstein DG-rings are local-Cohen-Macaulay. Our work is in a non-positive cohomological situation, allowing the Cohen-Macaulay condition to be introduced to derived algebraic geometry, but we also discuss extensions of it to non-negative DG-rings, which could lead to the concept of Cohen-Macaulayness in topology.

Original languageEnglish
Pages (from-to)6095-6138
Number of pages44
JournalTransactions of the American Mathematical Society
Volume373
Issue number9
DOIs
StatePublished - 1 Sep 2020
Externally publishedYes

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics

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