Open loci of graded modules

Rotthaus, Christel; Şega, Liana

doi:10.1090/S0002-9947-06-03876-1

Open loci of graded modules
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by Christel Rotthaus and Liana M. Şega PDF

Trans. Amer. Math. Soc. 358 (2006), 4959-4980 Request permission

Abstract:

Let $A=\bigoplus _{i\in \mathbb {N}}A_i$ be an excellent homogeneous Noetherian graded ring and let $M=\bigoplus _{n\in \mathbb {Z}}M_n$ be a finitely generated graded $A$-module. We consider $M$ as a module over $A_0$ and show that the $(S_k)$-loci of $M$ are open in $\operatorname {Spec}(A_0)$. In particular, the Cohen-Macaulay locus $U^0_{CM}=\{\mathfrak {p}\in \operatorname {Spec}(A_0) \mid M_\mathfrak {p} \; \mbox {is Cohen-Macaulay}\}$ is an open subset of $\operatorname {Spec}(A_0)$. We also show that the $(S_k)$-loci on the homogeneous parts $M_n$ of $M$ are eventually stable. As an application we obtain that for a finitely generated Cohen-Macaulay module $M$ over an excellent ring $A$ and for an ideal $I\subseteq A$ which is not contained in any minimal prime of $M$, the $(S_k)$-loci for the modules $M/I^nM$ are eventually stable.

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