The fundamental module of a normal local domain of dimension $2$

Abstract:

The fundamental module $E$ of a normal local domain $(R, \mathfrak {m})$ of dimension $2$ is defined by the nonsplit exact sequence $0 \to K \to E \to \mathfrak {m} \to 0$, where $K$ is the canonical module of $R$. We prove that, if $R$ is complete with $R/\mathfrak {m} \simeq \mathbb {C}$, then $E$ is decomposable if and only if $R$ is a cyclic quotient singularity. Various other properties of fundamental modules will be discussed.