Hilbert-Samuel functions of Cohen-Macaulay rings

Abstract:

Let $R$ be a local ring with a maximal ideal $\mathfrak {m}$. It is proved that in case $R$ is a Cohen-Macaulay (C.M.) ring and $\dim \mathfrak {m}/{\mathfrak {m}^2} - \dim R = 1$, then the multiplicity of $R$ and its dimension determine uniquely the Hilbert-Samuel function of $R$. As a corollary we obtain that the C.M. property is determined by the Hilbert-Samuel function in case $\dim \mathfrak {m}/{\mathfrak {m}^2} - \dim R = 1$. An example is given which shows that it is not so in case $\dim \mathfrak {m}/{\mathfrak {m}^2} - \dim R > 1$.