Injective DG-modules over non-positive DG-rings

Abstract

Let $A$ be an associative non-positive differential graded ring. In this paper we make a detailed study of a category $\mathrm{Inj}(A)$ of left DG-modules over $A$ which generalizes the category of injective modules over a ring. We give many characterizations of this category, generalizing the theory of injective modules, and prove a derived version of the Bass-Papp theorem: the category $\mathrm{Inj}(A)$ is closed in the derived category $D(A)$ under arbitrary direct sums if and only if the ring ${\mathrm{H}}^0(A)$ is left noetherian and for every $i<0$ the left ${\mathrm{H}}^0(A)$-module ${\mathrm{H}}^i(A)$ is finitely generated. Specializing further to the case of commutative noetherian DG-rings, we generalize the Matlis structure theory of injectives to this context. As an application, we obtain a concrete version of Grothendieck's local duality theorem over commutative noetherian local DG-rings.

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

References in corpus (2)

• Homological dimensions of local (co)homology over commutative DG-rings
• Adic reduction to the diagonal and a relation between cofiniteness and derived completion

Cited by in corpus (13)

• $t$-Structures with Grothendieck hearts via functor categories
• Completion and torsion over commutative DG rings
• The Cohen-Macaulay property in derived commutative algebra
• Koszul complexes over Cohen-Macaulay rings
• T-structures and twisted complexes on derived injectives
• On commutative differential graded algebras
• Sequence-regular commutative DG-rings
• Open loci results for commutative DG-rings
• A derived Gabriel-Popescu theorem for t-structures via derived injectives
• Mutation and torsion pairs
• Maximal Cohen-Macaulay DG-complexes
• Categorical properties of reduction functors over non-positive DG-rings
• G-dimensions for DG-modules over commutative DG-rings