The Cohen-Macaulay property in derived commutative algebra
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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.References
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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
- 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.
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 6095-6138
- MSC (2010): Primary 13C14, 13D45, 16E35, 16E45
- DOI: https://doi.org/10.1090/tran/8099
- MathSciNet review: 4155173