On pro-zero homomorphisms and sequences in local (co-)homology

Abstract

Let x_ denote a system of elements of a commutative ring R. For an R-module M we investigate when x_ is M-pro-regular resp. M-weakly pro-regular as generalizations of M-regular sequences. This is done in terms of Čech co-homology resp. homology, defined by Hi(Cˇx_⊗R⋅) resp. by Hi(RHomR(Cˇx_,⋅))≅Hi(HomR(Lx_,⋅)), where Cˇx_ denotes the Čech complex and Lx_ is a bounded free resolution of it as constructed in [17] resp. [16]. The property of x_ being M-pro-regular resp. M-weakly pro-regular follows by the vanishing of certain Čech co-homology resp. homology modules, which is related to completions. This extends previously work by Greenlees and May (see [5]) and Lipman et al. (see [1]). This contributes to a further understanding of Čech (co-)homology in the non-Noetherian case. As a technical tool we use one of Emmanouil's results (see [4]) about the inverse limits and its derived functor. As an application we prove a global variant of the results with an application to prisms in the sense of Bhatt and Scholze (see [3]).