On the monotonicity of Hilbert functions

Abstract

In this paper we show that a large class of one-dimensional Cohen–Macaulay local rings $(\mathcal A,\mathfrak m)$ 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.