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 language | English |
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Pages (from-to) | 6095-6138 |
Number of pages | 44 |
Journal | Transactions of the American Mathematical Society |
Volume | 373 |
Issue number | 9 |
DOIs | |
State | Published - 1 Sep 2020 |
Externally published | Yes |
ASJC Scopus subject areas
- General Mathematics
- Applied Mathematics