Almost split sequences and Zariski differentials

Abstract:

Let $R$ be a complete two-dimensional integrally closed analytic $k$-algebra. Associated with $R$ is the Auslander module $A$ from the fundamental sequence $0 \to {\omega _R} \to A \to R \to k \to 0$ and the module of Zariski differentials ${D_k}{(R)^{ * * }}$. We conjecture that these modules are isomorphic if and only if $R$ is graded. We prove this conjecture for (a) hypersurfaces $f = X_3^n + {\text {g}}({X_1},{X_2})$, (b) quotient singularities, and (c) $R$ graded Gorenstein.