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The Cohen-Macaulay property in derived commutative algebra
- Source :
- Transactions of the American Mathematical Society. 373:6095-6138
- Publication Year :
- 2020
- Publisher :
- American Mathematical Society (AMS), 2020.
-
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 J{\o}rgensen and resolving a recent conjecture of Minamoto. When these inequalities are equalities, we arrive to 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.<br />Comment: 41 pages, final version, to appear in Transactions of the AMS
- Subjects :
- Pure mathematics
Conjecture
Property (philosophy)
Mathematics::Commutative Algebra
Applied Mathematics
General Mathematics
010102 general mathematics
Dimension (graph theory)
13C14, 13D45, 16E45, 16E35
Local cohomology
Commutative Algebra (math.AC)
Mathematics - Commutative Algebra
01 natural sciences
Injective function
Mathematics - Algebraic Geometry
Derived algebraic geometry
FOS: Mathematics
Algebraic Topology (math.AT)
Mathematics - Algebraic Topology
0101 mathematics
Commutative algebra
Algebraic Geometry (math.AG)
Commutative property
Mathematics
Subjects
Details
- ISSN :
- 10886850 and 00029947
- Volume :
- 373
- Database :
- OpenAIRE
- Journal :
- Transactions of the American Mathematical Society
- Accession number :
- edsair.doi.dedup.....e382b6e84277160f4cdbda8fb1fcae80