Homological Dimensions of Local (Co)homology Over Commutative DG-rings

Abstract

AbstractLet 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 ɑ-torsion elements is also an injective A-module. Recently, de Jong proved a dual result: If F is a flat A-module, then the ɑ-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 ⊆ H0(A), we show that the local cohomology functor R associated with does not increase injective dimension. Dually, the derived -adic completion functor LΛ does not increase flat dimension.