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 $(A,\bar{\mathfrak{m}})$ 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.

28 pages, final version, to appear in Journal of Algebra

References in corpus (8)

• Injective DG-modules over non-positive DG-rings
• Completion and torsion over commutative DG rings
• The Cohen-Macaulay property in derived commutative algebra
• Koszul complexes over Cohen-Macaulay rings
• The twisted inverse image pseudofunctor over commutative DG rings and perfect base change
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Cited by in corpus (1)

• G-dimensions for DG-modules over commutative DG-rings