Homological dimensions of local (Co)Homology over commutative DG-rings

Let A be a commutative noetherian ring, let a ⊆ A be an ideal, and let I be an injective A-module. A basic result in the structure theory of injective modules states that the A-module Γa(I) consisting of a-torsion elements is also an injective A-module. Recently, de Jong proved a dual result: If F is a flat A-module, then the a-adic completion of F is also a flat A-module. In this paper we generalize these facts to commutative noetherian DG-rings: let A be a commutative non-positive DG-ring such that H⁰(A) is a noetherian ring and for each i < 0, the H⁰(A)-module Hⁱ(A) is finitely generated. Given an ideal a ⊆ H⁰(A), we show that the local cohomology functor RΓ_a associated with a does not increase injective dimension. Dually, the derived a-adic completion functor LΛ_a does not increase flat dimension.