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.