On the Monotonicity of Hilbert functions

TL;DR

This paper proves that for a broad class of one-dimensional Cohen-Macaulay local rings, the Hilbert function of any maximal Cohen-Macaulay module is non-decreasing, with specific examples including certain complete intersections and quotients of Cohen-Macaulay rings.

Contribution

It establishes the monotonicity of Hilbert functions for maximal Cohen-Macaulay modules over new classes of one-dimensional Cohen-Macaulay rings.

Findings

Hilbert functions are non-decreasing for these rings

Includes complete intersections with specific properties

Covers quotients of Cohen-Macaulay rings with pseudo-rational singularities

Abstract

In this paper we show that a large class of one-dimensional Cohen-Macaulay local rings $(A,\mathfrak{m})$ has the property that if $M$ is a maximal Cohen-Macaulay $A$-module then the Hilbert function of $M$ ( with respect to $\mathfrak{m}$) is non-decreasing. Examples include (1) Complete intersections $A = Q/(f,g)$ where $(Q,\mathfrak{n})$ is regular local of dimension three and $f \in \mathfrak{n}^2 \setminus \mathfrak{n}^3$. (2) One dimensional Cohen-Macaulay quotients of a two dimensional Cohen-Macaulay local ring with pseudo-rational singularity.