Quasi-pure resolutions and some lower bounds of Hilbert coefficients of Cohen-Macaulay modules

TL;DR

This paper investigates the properties of Cohen-Macaulay modules over Gorenstein local rings, establishing conditions under which associated graded modules are Cohen-Macaulay and providing lower bounds for Hilbert coefficients in various algebraic contexts.

Contribution

It proves that quasi-pure resolutions imply Cohen-Macaulayness of associated graded modules and derives new lower bounds for Hilbert coefficients in specific Cohen-Macaulay module settings.

Findings

G(M) is Cohen-Macaulay if G(M) has a quasi-pure resolution over a regular A.

Lower bounds for e_0(M) and e_1(M) are established when G(A) is Cohen-Macaulay and M has finite projective dimension.

Lower bounds on Hilbert coefficients are given for modules over strict complete intersections with certain codimension and complexity.

Abstract

Let $(A,\mathfrak{m})$ be a Gorenstein local ring and let $M$ be a finitely generated Cohen Macaulay $A$ module. Let $G(A)=\bigoplus_{n\geq 0}\mathfrak{m}^n/\mathfrak{m}^{n+1}$ be the associated graded ring of $A$ and $G(M)=\bigoplus_{n\geq 0}\mathfrak{m}^nM/\mathfrak{m}^{n+1}M$ be the associated graded module of $M$. If $A$ is regular and if $G(M)$ has a quasi-pure resolution then we show that $G(M)$ is Cohen-Macaulay. If $G(A)$ is Cohen-Macaulay and if $M$ has finite projective dimension then we give lower bounds on $e_0(M)$ and $e_1(M)$. Finally let $A = Q/(f_1, \ldots, f_c)$ be a strict complete intersection with $\text{ord}(f_i) = s$ for all $i$. Let $M$ be an Cohen-Macaulay module with $\text{cx}_A(M) = r < c$. We give lower bounds on $e_0(M)$ and $e_1(M)$.