Proregular sequences, local cohomology, and completion

Abstract

As a certain generalization of regular sequences there is an investigation of weakly proregular sequences. Let M denote an arbitrary R-module. As the main result it is shown that a system of elements x- with bounded torsion is a weakly proregular sequence if and only if the cohomology of the Čech complex Čx- ⊗ M is naturally isomorphic to the local cohomology modules Hscript a signi (M) and if and only if the homology of the co-Čech complex RHom(Čx-, M) is naturally isomorphic to LiΛscript a sign(M), the left derived functors of the script a sign-adic completion, where a denotes the ideal generated by the elements x-. This extends results known in the case of R a Noetherian ring, where any system of elements forms a weakly proregular sequence of bounded torsion. Moreover, these statements correct results previously known in the literature for proregular sequences.