Proregular sequences, local cohomology, and completion - Schenzel, Peter (Mathematica Scandinavica, 2003)
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 C_x M is naturally isomorphic to the local cohomology modules H_a^i(M) and if and only if the homology of the co-Čech complex RHom (C_x, M) is naturally isomorphic to L_i Λ^a(M), the left derived functors of the a-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.