On the formal grade of finitely generated modules over local rings

Details On the formal grade of finitely generated modules over local rings On the formal grade of finitely generated modules over local rings

2008-08-05

arxiv.org/abs/0808.0653v4

Mathematics

Commutative Algebra

This paper has been withdrawn by the authors, because it has been included as a section in arXiv:0807.5042

Scientific paper

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)\neq 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.

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