# Ulrich module

A maximal Cohen-Macaulay module $M$ over a Noetherian local ring $(R,\mathfrak{m},k)$ is Ulrich if $e(M)=\mu(M)$, where $e(M)$ is the Hilbert-Samuel multiplicity of $M$ and $\mu(M)$ is the minimal number of generators of $M$. When $M$ is a maximal Cohen-Macaulay module and $\mathfrak{m}$ has a minimal reduction $I$ generated by a system of parameters, $M$ is Ulrich if and only if $\mathfrak{m}M=IM$.

Title Ulrich module UlrichModule 2013-03-22 18:13:38 2013-03-22 18:13:38 yshen (21076) yshen (21076) 8 yshen (21076) Definition msc 13C14