On the index of reducibility in Noetherian modules

TL;DR

This paper investigates the index of reducibility in Noetherian modules, providing formulas, conditions for additivity, polynomial behavior, and characterizations in Cohen-Macaulay modules.

Contribution

It offers new formulas and characterizations for the index of reducibility, including polynomial bounds and Cohen-Macaulay module criteria.

Findings

Explicit formula for ir_M(N) in terms of associated primes and socles.

Additivity of ir_M(N) for primary decompositions with maximal embedded components.

Existence of a polynomial describing ir_M(I^n M) for large n.

Abstract

Let $M$ be a finitely generated module over a Noetherian ring $R$ and $N$ a submodule. The index of reducibility ir$_M(N)$ is the number of irreducible submodules that appear in an irredundant irreducible decomposition of $N$ (this number is well defined by a classical result of Emmy Noether). Then the main results of this paper are: (1) $\mathrm{ir}_M(N) = \sum_{{\frak p} \in \mathrm{Ass}_R(M/N)} \dim_{k(\frak p)} \mathrm{Soc}(M/N)_{\frak p} $; (2) For an irredundant primary decomposition of $N = Q_1 \cap \cdots \cap Q_n$, where $Q_i$ is $\frak p_i$-primary, then $\mathrm{ir}_M(N) = \mathrm{ir}_M(Q_1) + \cdots + \mathrm{ir}_M(Q_n)$ if and only if $Q_i$ is a $\frak p_i$-maximal embedded component of $N$ for all embedded associated prime ideals $\frak p_i$ of $N$; (3) For an ideal $I$ of $R$ there exists a polynomial $\mathrm{Ir}_{M,I}(n)$ such that $\mathrm{Ir}_{M,I}(n)=\mathrm{ir}_M(I^nM)$ for $n\gg 0$. Moreover, $\mathrm{bight}_M(I)-1\le \deg(\mathrm{Ir}_{M,I}(n))\le \ell_M(I)-1$; (4) If $(R, \frak m)$ is local, $M$ is Cohen-Macaulay if and only if there exist an integer $l$ and a parameter ideal $\frak q$ of $M$ contained in $\frak m^l$ such that $\mathrm{ir}_M({\frak q}M)=\dim_k\mathrm{Soc}(H^d_{\frak m}(M))$, where $d=\dim M$.