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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The Cohen-Macaulay property in derived commutative algebra


Author: Liran Shaul
Journal: Trans. Amer. Math. Soc. 373 (2020), 6095-6138
MSC (2010): Primary 13C14, 13D45, 16E35, 16E45
DOI: https://doi.org/10.1090/tran/8099
Published electronically: June 24, 2020
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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.


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Additional Information

Liran Shaul
Affiliation: Department of Algebra, Faculty of Mathematics and Physics, Charles University in Prague, Sokolovská 83, 186 75 Praha, Czech Republic
MR Author ID: 1050601
Email: shaul@karlin.mff.cuni.cz

DOI: https://doi.org/10.1090/tran/8099
Received by editor(s): June 10, 2019
Received by editor(s) in revised form: November 27, 2019, and December 10, 2019
Published electronically: June 24, 2020
Additional Notes: The author was partially supported by the Israel Science Foundation (grant no. 1346/15). This work has been supported by Charles University Research Centre program No.UNCE/SCI/022.
Article copyright: © Copyright 2020 American Mathematical Society