  By extending some basic results of Grothendieck and Foxby about local
cohomology to commutative DG-rings, we prove new amplitude inequalities about
finite DG-modules of finite injective dimension over commutative local
DG-rings, complementing results of J{\o}rgensen and resolving a recent
conjecture of Minamoto. When these inequalities are equalities, we arrive to
the notion of a local-Cohen-Macaulay DG-ring. We make a detailed study of this
notion, showing that much of the classical theory of Cohen-Macaulay rings and
modules can be generalized to the derived setting, and that there are many
natural examples of local-Cohen-Macaulay DG-rings. In particular, local
Gorenstein DG-rings are local-Cohen-Macaulay. Our work is in a non-positive
cohomological situation, allowing the Cohen-Macaulay condition to be introduced
to derived algebraic geometry, but we also discuss extensions of it to
non-negative DG-rings, which could lead to the concept of Cohen-Macaulayness in
topology.
