Almost split sequences and Zariski differentials

Abstract

Let R R be a complete two-dimensional integrally closed analytic k k -algebra. Associated with R R is the Auslander module A A from the fundamental sequence 0 → ω R → A → R → k → 0 0 \to {\omega _R} \to A \to R \to k \to 0 and the module of Zariski differentials D k ( R ) ∗ ∗ {D_k}{(R)^{ * * }} . We conjecture that these modules are isomorphic if and only if R R is graded. We prove this conjecture for (a) hypersurfaces f = X 3 n + g ( X 1 , X 2 ) f = X_3^n + {\text {g}}({X_1},{X_2}) , (b) quotient singularities, and (c) R R graded Gorenstein.