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

Let = x₁,…,xᵣ denote a system of elements of a commutative ring R. For an R-module M we investigate when is M-pro-regular resp. M-weakly pro-regular as generalizations of M-regular sequences. This is done in terms of {C}ech co-homology resp. homology, defined by Hⁱ(C_ ⊗R ·) resp. by Hᵢ(R R(C_,·)) Hᵢ(R(L_,·)), where C_ denotes the {C}ech complex and L_ is a bounded free resolution of it as constructed in [17] resp. [16]. The property of being M-pro-regular resp. M-weakly pro-regular follows by the vanishing of certain {C}ech 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 {C}ech (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]).