On the formal grade of finitely generated modules over local rings

2008/08/05 by Mohsen Asgharzadeh, Asgharzadeh, Mohsen, Kamran Divaani-Aazar +1
Mathematics · #Commutative Algebra and Its Applications #Rings, Modules, and Algebras #Algebraic structures and combinatorial models

paper · pdf · doi:10.48550/arxiv.0808.0653

Abstract

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_nHⁱ_\fm(M/\faⁿ M)≠ 0). We show that \fgrade(\fa,M)≥ \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.

Citations

• Cohen-Macaulay Rings, Invariant Theory, and the Generic Perfection of Determinantal Loci

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