On a new invariant of finitely generated modules over local rings

TL;DR

This paper introduces a new invariant for finitely generated modules over local rings, showing its independence from parameter choices and exploring its relation to module filtrations and Cohen-Macaulay properties.

Contribution

It defines a novel invariant p_F(M) for modules with filtrations, proving its independence from parameter systems and relating it to polynomial types and Cohen-Macaulay loci.

Findings

p_F(M) is independent of the choice of good systems of parameters.

The paper establishes relations between p_F(M) and polynomial types of module quotients.

Connections are made between p_F(M) and the dimension of the non-sequentially Cohen-Macaulay locus.

Abstract

Let $M$ be a finitely generated module on a local ring $R$ and $\F: M_0\subset M_1\subset...\subset M_t=M$ a filtration of submodules of $M$ such that $ d_o<d_1< ... <d_t=d$, where $d_i=\dim M_i$. This paper is concerned with a non-negative integer $p_\mathcal F(M)$ which is defined as the least degree of all polynomials in $n_1, ..., n_d$ bounding above the function $$\ell(M/(x_1^{n_1}, ..., x_d^{n_d})M)-\sum_{i=0}^tn_1...n_{d_i}e(x_1,..., x_{d_i};M_i).$$ We prove that $p_\mathcal F(M)$ is independent of the choices of good systems of parameters $\underline x=x_1, ..., x_d$. When $\F$ is the dimension filtration of $M$ we also present some relations between $p_\F(M)$ and the polynomial type of each $M_i/M_{i-1}$ and the dimension of the non-sequentially Cohen-Macaulay locus of $M$.