Über die assoziierten Primideale der Vervollständigung

Abstract

Let $(R,\my)$ be a noetherian local ring and let $M$ be an $R$-module such that $\bigcap\limits_{n\geq 1} \my^n M=0.$ Let $\hat{M}$ be the completion of $M$. We show that Ass$(\hat{M})=$ Koatt$(M)$ holds in the following three cases: if $\dim (R)\le 1,$ if $\hat{M}$ as $R$-module is flat, or if $M$ is the direct sum of $R$-modules which are finitely generated. If $M$ is pure in $\hat{M}$ then at least Ass$(\hat{M})\subset$ Koatt$(M)$ holds. If the conjecture by A.-M.Simon on complete $R$-modules is valid then one has Koatt$(M)\subset$ Ass$(\hat{M}).$

14 pages