Let fa be an ideal of a local ring (R,fm) and M a finitely generated R-module. This paper concerns 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). We show that fgr
ade(fa,M)geq depth M-cd_{fa}(M), and as a result, we establish a new characterization of Cohen-Macaulay modules. As an application of this characterization, we show that if M is Cohen-Macaulay and L a pure submodule of M with the same support as M, then fgrade(fa,L)=fgrade(fa,M). Also, we give a generalization of the Hochster-Eagon result on Cohen-Macaulayness of invariant rings.
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 propertie
s 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
on of M with respect to fa. When (R,fm) is local and r:=f_{fa}(M) is less than f_{fa}^{fm}(M), the fm-finiteness dimension of M relative to fa, we prove that H^r_{fa}(M) is not Artinian, and so the filter depth of fa on M doesnt exceeds f_{fa}(M). Also, we show that if M has finite dimension and H^i_{fa}(M) is Artinian for all i>t, where t is a given positive integer, then H^t_{fa}(M)/fa H^t_{fa}(M) is Artinian. It immediately implies that if q:=q_{fa}(M)>0, then H^q_{fa}(M) is not finitely generated, and so f_{fa}(M)leq q_{fa}(M).
