On Solutions for Derivations of a Noetherian<i>k</i>-algebra and Local Simplicity

Abstract

We introduce a general notion of solution for a Noetherian differential k-algebra and study its relationship with simplicity, where k is an algebraically closed field; then we analyze conditions under which such solutions may exist and be unique, with special emphasis in the cases of k-algebras of finite type and formal series rings over k. Using that notion we generalize a criterion for simplicity due to Brumatti–Lequain–Levcovitz and give a geometric characterization of that; as an application we give a new proof of a classification theorem for local simplicity due to Hart and obtain a general result for simplicity of formal series rings over k.