On associated graded modules of maximal Cohen-Macaulay modules over hypersurface rings

TL;DR

This paper investigates the depth properties of associated graded modules of maximal Cohen-Macaulay modules over hypersurface rings, establishing new bounds and conditions that advance understanding in this area.

Contribution

It provides the first systematic study of the depth of associated graded modules of MCM modules over hypersurface rings, deriving new bounds under specific conditions.

Findings

Depth of G(M) is at least d-1 when e(M)=μ(M)i(M)+1.

Depth of G(M) is at least d-μ(M)+1 when e(A)=3 and μ(M) is 2 or 3.

New bounds for the depth of associated graded modules in hypersurface rings.

Abstract

Let $A=Q/(f)$ where $(Q,\mathfrak{n})$ be a complete regular local ring of dimension $d+1$, $f\in \mathfrak{n}^i\setminus\mathfrak{n}^{i+1}$ for some $i\geq 2$ and $M$ an MCM $A-$module with $e(M)=\mu(M)i(M)+1$ then we prove that depth $G(M)\geq d-1$. If $(A,\mathfrak{m})$ is a complete hypersurface ring of dimension $d$ with infinite residue field and $e(A)=3$, let $M$ be an MCM $A$-module with $\mu(M)=2$ or $3$ then we prove that depth $G(M)\geq d-\mu(M)+1$. Our paper is the first systematic study of depth of associated graded modules of MCM modules over hypersurface rings.