Sequence-regular commutative DG-rings

TL;DR

This paper introduces a new class of commutative noetherian DG-rings called sequence-regular DG-rings, generalizing regular local rings, and explores their properties and applications in derived algebraic geometry.

Contribution

The paper defines sequence-regular DG-rings, proves a generalized Auslander-Buchsbaum-Serre theorem, and demonstrates their ubiquity in derived algebraic varieties over perfect fields.

Findings

Generalization of regular local rings to DG-rings.

Proof of a generalized Auslander-Buchsbaum-Serre theorem.

Ubiquity of sequence-regular DG-rings in derived algebraic geometry.

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 $(A,\bar{\mathfrak{m}})$ such that the maximal ideal $\bar{\mathfrak{m}} \subseteq \mathrm{H}^0(A)$ 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.