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Sequence-regular commutative DG-rings

Liran Shaul

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.

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 such that the maximal ideal can be generated by an -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.

Paper Structure

This paper contains 13 sections, 38 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Commutative DG-rings
  4. Localization
  5. Regular sequences and Koszul complexes
  6. Dualizing DG-modules
  7. Cohen–Macaulay DG-rings
  8. Local-Cohen–Macaulay DG-rings with constant amplitude
  9. Sequence-regular DG-rings
  10. Behavior of sequence-regularity with respect to standard operations
  11. The residue DG-field of a sequence-regular DG-ring
  12. Generic sequence-regularity
  13. Special commutative DG-rings

Key Result

Theorem 1

Let $(A,\bar{\mathfrak{m}})$ be a sequence-regular noetherian local DG-ring. Then for any prime $\bar{\mathfrak{p}} \in \operatorname{Spec}(\mathrm{H}^0(A))$, the localization $A_{\bar{\mathfrak{p}}}$ is also a sequence-regular DG-ring.

Theorems & Definitions (83)

  • Definition
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 73 more