On associated graded modules having a pure resolution

TL;DR

This paper characterizes when the associated graded module of a Cohen-Macaulay module over a power series ring has a pure resolution, providing a precise criterion in algebraic terms.

Contribution

It establishes a necessary and sufficient condition for the associated graded module to have a pure resolution, advancing understanding of module resolutions in algebraic geometry.

Findings

Provides a criterion for pure resolutions of associated graded modules

Connects Cohen-Macaulay modules with pure resolutions over polynomial rings

Enhances the theoretical framework for module resolution analysis

Abstract

Let $A = K[[X_1,\cdots,X_n]]$ and let $\mathfrak{m} = (X_1,\cdots,X_n)$. Let $M$ be a Cohen-Macaulay $A$-module of codimension $p$. In this paper we give a necessary and sufficient condition for the associated graded module $G_{\mathfrak{m}}(M)$ to have a pure resolution over the polynomial ring $G_{\mathfrak{m}}(A) \cong K[X_1,\cdots,X_n]$.