On pro-zero homomorphisms and sequences in local (co-)homology (Q6541321)

Let \(R\) be a commutative ring, and \(\underline{a}= a_{1},\dots,a_{n}\in R\). Given an \(R\)-module \(M\), the author investigates when the sequence \(\underline{a}\) is \(M\)-proregular or \(M\)-weakly pro-regular as generalizations of \(M\)-regularity. This is done in terms of Čech homology \(H_{i}\left(\operatorname{RHom}_{R}(\check{C}_{\underline{a}},-)\right)\) and Čech cohomology \(H_{-i}\left(\check{C}_{\underline{a}}\otimes_{R}^{\textrm{L}}-\right)\) functors where \(\check{C}_{\underline{a}}\) denotes the Čech complex. The property of \(\underline{a}\) being \(M\)-proregular or \(M\)-weakly pro-regular follows from the vanishing of certain Čech cohomology or homology modules which is related to completions. This extends the previous work of Greenlees, May, and Lipman, and also contributes to a further understanding of Čech (co-)homology in the non-noetherian setting. As a technical tool, the author uses one of Emmanouil's results on inverse limits and their derived functors. Furthermore, he proves a global variant of his results with an application to prisms in the sense of Bhatt and Scholze.