Sequence-regular commutative DG-rings

Abstract

We introduce a new class of commutative noetherian DG-rings which generalizes the class of regular local rings. These are defined to be local DG-rings (A, m) such that the maximal ideal m ⊆ H 0 (A) can be generated by an A-regular sequence. We call these DG-rings sequence-regular DG-rings, and make a detailed study of them. Using methods of Cohen-Macaulay differential graded algebra, we prove that the Auslander-Buchsbaum-Serre theorem about localization generalizes to this setting. This allows us to define global sequence-regular DG-rings, and to introduce this regularity condition to derived algebraic geometry. It is shown that these DG-rings share many properties of classical regular local rings, and in particular we are able to construct canonical residue DG-fields in this context. Finally, we show that sequence-regular DG-rings are ubiquitous, and in particular, any eventually coconnective derived algebraic variety over a perfect field is generically sequence-regular.

Finitistic dimensions over commutative DG-rings

“…We warn the reader that unlike in the case of rings, in general local-Cohen-Macaulay does not imply Cohen-Macaulay due to localization causing a drop in amplitude, see [23,Example 8.3]. However, when the spectrum of H 0 (A) is irreducible and H 0 (A) is catenary, the two definitions agree by combining [23,Corollary 8.7] and [25,Theorem 2.5]. Proposition 5.2.…”