Open loci results for commutative DG-rings

Abstract

Given a commutative noetherian non-positive DG-ring with bounded cohomology which has a dualizing DG-module, we study its regular, Gorenstein and Cohen-Macaulay loci. We give a sufficient condition for the regular locus to be open, and show that the Gorenstein locus is always open. However, both of these loci are often empty: we show that no matter how nice ${\mathrm{H}}^0(A)$ is, there are examples where the Gorenstein locus of $A$ is empty. We then show that the Cohen-Macaulay locus of a commutative noetherian DG-ring with bounded cohomology which has a dualizing DG-module always contains a dense open set. Our results imply that under mild hypothesis, eventually coconnective locally noetherian derived schemes are generically Cohen-Macaulay, but that even in very nice cases, they need not be generically Gorenstein.

10 pages, final version, to appear in Journal of Pure and Applied Algebra

References in corpus (4)

• Derived Categories
• Injective DG-modules over non-positive DG-rings
• The Cohen-Macaulay property in derived commutative algebra
• On commutative differential graded algebras

Cited by in corpus (1)

• Sequence-regular commutative DG-rings