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

TL;DR

This paper systematically investigates the depth of associated graded modules of maximal Cohen-Macaulay modules over hypersurface rings, providing new bounds and estimates based on module invariants and specific ring conditions.

Contribution

It offers the first comprehensive analysis of the depth of associated graded modules of MCM modules over hypersurface rings, establishing new bounds and estimates.

Findings

Depth bounds for MCM modules with specific multiplicity and minimal number of generators.

Depth estimates when $e(M)=rac{rac{M}{rac{M}{i(M)}}+1$.

Explicit estimates for hypersurface rings of the form $Q/(f)$.

Abstract

Let $(A,\mathfrak{m})$ be a hypersurface ring with dimension $d$, and $M$ a MCM $A-$module with red$(M)\leq 2$ and $\mu(M)=2$ or $3$ then we have proved that depth $G(M)\geq d-\mu(M)+1$. If $e(A)=3$ and $\mu(M)=4$ then in this case we have proved that depth$G(M)\geq d-3$. Next we consider the case when $e(M)=\mu(M)i(M)+1$ and prove that depth $G(M)\geq d-1$. When $A = Q/(f)$ where $Q = k[[X_1,\cdots, X_{d+1}]]$ then we give estimates for $\depth G(M)$ in terms of a minimal presentation of $M$. Our paper is the first systematic study of depth of associated graded modules of MCM modules over hypersurface rings.