Homological dimensions of local (co)homology over commutative DG-rings

Let $A$ be a commutative noetherian ring, let $\mathfrak{a}\subseteq A$ 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 ${\mathrm{\Gamma }}_{\mathfrak{a}}(I)$ consisting of $\mathfrak{a}$-torsion elements is also an injective $A$-module. Recently, de Jong proved a dual result: If $F$ is a flat $A$-module, then the $\mathfrak{a}$-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 ${\mathrm{H}}^0(A)$ is a noetherian ring, and for each $i<0$, the ${\mathrm{H}}^0(A)$-module ${\mathrm{H}}^i(A)$ is finitely generated. Given an ideal $\bar{\mathfrak{a}}\subseteq {\mathrm{H}}^0(A)$, we show that the local cohomology functor $\mathrm{R}{\mathrm{\Gamma }}_{\bar{\mathfrak{a}}}$ associated to $\bar{\mathfrak{a}}$ does not increase injective dimension. Dually, the derived $\bar{\mathfrak{a}}$-adic completion functor $\mathrm{L}{\mathrm{\Lambda }}_{\bar{\mathfrak{a}}}$ does not increase flat dimension.