\"Uber die assoziierten Primideale der Vervollst\"andigung

TL;DR

This paper investigates the relationship between associated primes of the completion of a module and its coassociated primes over noetherian local rings, establishing equalities in specific cases and proposing conditions for inclusion.

Contribution

It proves the equality Ass$(\hat{M})$ = Koatt$(M)$ in certain cases and explores conditions under which these sets are contained within each other, advancing understanding of prime ideal associations.

Findings

Equality holds if $ ext{dim}(R) extless=1$

Equality holds if $\hat{M}$ is flat as an $R$-module

If $M$ is a direct sum of finitely generated modules, the equality holds

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 $\dim(R)\leq 1,$ 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$(\hat{M}).$