Ali Soleyman Jahan: Easy proofs of some well known facts via cleanness, p.237-243

Abstract:

We give easy proofs for some well known facts by using some basic property of cleanness. We show that if (R, m) is a Noetherian local ring and $M$ is a finitely generated almost clean $R$-module with the property that $R/P$ is Cohen-Macaulay for all P ∈ Ass(M), then depth(M)=min{dim(R/P) : p ∈ Ass(M)}. Using this fact we show that if $M$ is a finitely generated clean $R$-module such that $R/P$ is Cohen-Macaulay and $\dim(M)=\dim(R/P)$ for all minimal prime ideals of $M$, then $M$ is Cohen-Macaulay. This implies the well known fact that a pure shellable simplicial complex is Cohen-Macaulay.

Key Words: Prime filtration, Shellable simplicial complex, Monomial ideals, Clean and pretty clean modules.

2000 Mathematics Subject Classification: Primary: 13C13
Secondary: 13F55, 13F20.

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