Asymptotic Behaviour of Parameter Ideals in Generalized Cohen-Macaulay Modules

TL;DR

This paper investigates the asymptotic properties of parameter ideals in generalized Cohen-Macaulay modules, providing affirmative answers to longstanding open questions about their invariance and algebraic relations in high powers.

Contribution

It proves that for large powers of the maximal ideal, the index of reducibility is independent of the parameter ideal and the relation $I^2 = q I$ holds, resolving two open questions.

Findings

The index of reducibility stabilizes for large powers of the maximal ideal.

The equality $I^2 = q I$ holds for parameter ideals in high powers.

Affirmative answers to open questions by Rogers and Goto-Sakurai.

Abstract

The purpose of this paper is to give affirmative answers to two open questions as follows. Let $(R, \m)$ be a generalized Cohen-Macaulay Noetherian local ring. Both questions, the first question was raised by M. Rogers \cite {R} and the second one is due to S. Goto and H. Sakurai \cite {GS1}, ask whether for every parameter ideal $\q$ contained in a high enough power of the maximal ideal $\m $ the following statements are true: (1) The index of reducibility $N_R(\q;R)$ is independent of the choice of $\q$; and (2) $I^2=\q I$, where $I=\q:_R\m$.