Injective DG-modules over non-positive DG-rings

Let A be an associative non-positive differential graded ring. In this paper we make a detailed study of a category 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 Inj(A) is closed in the derived category D(A) under arbitrary direct sums if and only if the ring H⁰(A) is left noetherian and for every i<0 the left H⁰(A)-module Hⁱ(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.