Totalseparierte Moduln

TL;DR

This paper studies totally separated modules over noetherian local rings, exploring conditions for purity, the relationship with completions, and properties of the $rak{m}$-adic closure, including radicality and prime ideals.

Contribution

It establishes new criteria for total separation of modules and their completions, linking purity, essential extensions, and the $rak{m}$-adic closure in module theory.

Findings

Total separation ensures purity in the completion under certain conditions.

Criteria for the $rak{m}$-adic closure to be radical are provided.

The coassociated primes of the $rak{m}$-adic closure are characterized.

Abstract

Let $(R, \mathfrak{m})$ be a noetherian local ring, $M$ a separated $R$-module (i.e. $\bigcap\limits_{n\geq 1}\mathfrak{m}^n M = 0$) and $\widehat{M} = \lim\limits_{\leftarrow} M/\mathfrak{m}^n M$ its completion. Generally, $M$ is not pure in $\widehat{M}$ and $\widehat{M}$ is not pure-injective. But if $M$ is totally separated, i.e. $X\underset{R}{\otimes} M$ is separated for all finitely generated $R$-modules $X$, the situation improves: In this case, $M$ is pure in $\widehat{M}$ and, under additional conditions, $\widehat{M}$ is even pure-injective, e.g. if $M\cong X^{(I)}$ holds with $X$ finitely generated or $M \cong\coprod_{i=1}^{\infty} R/\mathfrak{m}^i$. In section 2, we investigate the question under which conditions both $M$ and $\widehat{M}$ are totally separated and establish a close connection to the class of strictly pure-essential extensions. In section 3, we replace the completion $\widehat{M}$ in the case $M = \coprod_{i\in I}M_i$ with the $\mathfrak{m}$-adic closure $A$ of $M$ in $P = \prod_{i\in I} M_i$, i.e. with $A = \bigcap_{n \geq 1}(M + \mathfrak{m}^n P)$. We give criteria so that $A/M$ is radical and show that this always holds in the countable case $M = \coprod_{i=1}^{\infty} M_i$. Finally, we deal with the case that $A$ is even totally separated and additionally determine the coassociated prime ideals of $A/M$.