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Let $fa$ be an ideal of a local ring $(R,fm)$ and $M$ a finitely generated $R$-module. We investigate the structure of the formal local cohomology modules ${vpl}_nH^i_{fm}(M/fa^n M)$, $igeq 0$. We prove several results concerning finiteness properties of formal local cohomology modules which indicate that these modules behave very similar to local cohomology modules. Among other things, we prove that if $dim Rleq 2$ or either $fa$ is principal or $dim R/faleq 1$, then $Tor_j^R(R/fa,{vpl}_nH^i_{fm}(M/fa^n M))$ is Artinian for all $i$ and $j$. Also, we examine the notion $fgrade(fa,M)$, the formal grade of $M$ with respect to $fa$ (i.e. the least integer $i$ such that ${vpl}_nH^i_{fm}(M/fa^n M) eq 0$). As applications, we establish a criterion for Cohen-Macaulayness of $M$, and also we provide an upper bound for cohomological dimension of $M$ with respect to $fa$.
Let fa be an ideal of a commutative Noetherian ring R and M a finitely generated R-module. We explore the behavior of the two notions f_{fa}(M), the finiteness dimension of M with respect to fa, and, its dual notion q_{fa}(M), the Artinianess dimensi
Our goal is to determine when the trivial extensions of commutative rings by modules are Cohen-Macaulay in the sense of Hamilton and Marley. For this purpose, we provide a generalization of the concept of Cohen-Macaulayness of rings to modules.
Let $frak a$ be an ideal of a commutative noetherian ring $R$ with unity and $M$ an $R$-module supported at $V(fa)$. Let $n$ be the supermum of the integers $i$ for which $H^{fa}_i(M) eq 0$. We show that $M$ is $fa$-cofinite if and only if the $R$-mo
Let $A$ be a commutative Noetherian ring containing a field $K$ of characteristic zero and let $R= A[X_1, ldots, X_m]$. Consider $R$ as standard graded with $deg A=0$ and $deg X_i=1$ for all $i$. We present a few results about the behavior of the gra
We study the singularities of integral models of Shimura varieties and moduli stacks of shtukas with parahoric level structure. More generally our results apply to the Pappas-Zhu and Levin mixed characteristic parahoric local models, and to their equ