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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 fgrade(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.
It is proved that the minimal free resolution of a module M over a Gorenstein local ring R is eventually periodic if, and only if, the class of M is torsion in a certain Z[t,t^{-1}]-module associated to R. This module, denoted J(R), is the free Z[t,t
Let $A$ be a regular 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 $i$. Let $G$ be a finite subgroup of $GL_m(A)$. Let $G$ act linearly o
Let $mathbf{k}$ be a field of arbitrary characteristic, let $Lambda$ be a finite dimensional $mathbf{k}$-algebra, and let $V$ be a finitely generated $Lambda$-module. F. M. Bleher and the third author previously proved that $V$ has a well-defined ver
Numerical invariants of a minimal free resolution of a module $M$ over a regular local ring $(R, )$ can be studied by taking advantage of the rich literature on the graded case. The key is to fix suitable $ $-stable filtrations ${mathbb M} $ of $M $
It is proven that each indecomposable injective module over a valuation domain $R$ is polyserial if and only if each maximal immediate extension $widehat{R}$ of $R$ is of finite rank over the completion $widetilde{R}$ of $R$ in the $R$-topology. In t