Open loci of graded modules

Details Open loci of graded modules Open loci of graded modules

2004-03-23

arxiv.org/abs/math/0403399v1

Mathematics

Commutative Algebra

22 pages

Scientific paper

Let $A=\oplus_{i\in \nn}A_i$ be an excellent homogeneous Noetherian graded ring and let $M=\oplus_{n\in \zz}M_n$ 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 $\Spec(A_0)$. In particular, the Cohen-Macaulay locus $U^0_{CM}=\{\p\in \Spec(A_0) \mid M_\p {is Cohen-Macaulay}\}$ is an open subset of $\Spec(A_0)$. 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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