\"Uber die von einem Ideal $I \subset R$ erzeugten $R$-Moduln III

TL;DR

This paper investigates modules generated by ideals in local rings, exploring conditions for excellence, properties of trace modules, and dual constructions, with significant results on when certain equalities hold and their implications.

Contribution

It provides new criteria for modules to be excellent, characterizes trace modules for prime ideals, and links dual constructions to classical module properties in local rings.

Findings

Modules with vanishing Ext satisfy IM = gamma_I(M)

For prime ideals, gamma_p(R) equals p iff R_p is not a DVR

Gamma_m^n(R) equals the first neighborhood ring for almost all n

Abstract

Let $(R, \mathfrak m)$ be a commutative noetherian local ring and $I$ an ideal of $R$. For every $R$-module $M$, $\gamma_I(M) = \sum\{ \operatorname{Bi} f \,|\, f \in \operatorname{Hom}_R(I,M)\}$ is called the trace of $I$ in $M$. It is easy to see that $\operatorname{Ext}_R^1(R/I,M) = 0$ always implies $IM = \gamma_I(M)$. If the second condition holds for all ideals $I$ of $R$, we say that $M$ is excellent. In part 1, we show a number of conditions for these modules, which are well-known for injective modules. In the second part, we examine the special case $M = R$. In particular, we show that for every prime ideal $\mathfrak{p}$ the equality $\mathfrak{p} = \gamma_{\mathfrak{p}}(R)$ holds iff $R_{\mathfrak{p}}$ is not a discrete valuation ring. From the results by Matlis (1973) about 1-dimensional local CM-rings and with the help of the first neighborhood ring $\Lambda$, it follows immediately that $\gamma_{\mathfrak{m}^n} (R) = \Lambda^{-1}$ for almost all $n \geq 1$. In the third part, we examine the dual construction $\kappa_I(M) = \bigcap \{ \operatorname{Ke} f \,|\, f\in \operatorname{Hom}_R(M,I^\circ) \}$ and reduce the main results about $\operatorname{Tor}_1^R(M, R/I) = 0$ and $\kappa_I(M) = M[I]$ to part 1 by considering the Matlis dual $M^\circ = \operatorname{Hom}_R(M, E)$ and the equalities $\gamma_I(M^\circ) = \operatorname{Ann}_{M^\circ}(\kappa_I(M))$, $\kappa_I(M^\circ) = \operatorname{Ann}_{M^\circ}(\gamma_I(M))$.