Isomorphisms of Rings of Differential Operators on Curves

Abstract

Let X and Y be nonisomorphic irreducible affine algebraic curves over the complex numbers C. Let D{X) and D{Y) be their rings of differential operators (see

On the Isomorphism Problem for Rings of Differential Operators Over Affine Curves

“…In the case of isomorphism, we define an explicit isomorphism. In particular, we make explicit isomorphisms announced in Letzter (1992) and Perkins (1991). Notice that in case k is an algebraically closed field of characteristic zero, the class of algebras of differential operators (X(b)) is a very important one, since any k-algebra Morita equivalent to A 1 is isomorphic to some (X(b)) in Kouakou (2003).…”

“…This problem is a particular case of the general problem of Stafford (1987) on isomorphisms between two k-algebras and both Morita equivalent to A 1 . In this paper, we study affine algebraic curves X (b) introduced by Letzter (1992) and Perkins (1991) and their algebra of differential operators (X(b)). Due to the resolution of the problem above, we find the condition to have an isomorphism between two such algebras of differential operators.…”

“…Dans le cas d'isomorphisme, nous donnons explicitement un isomorphisme. En particulier, nous explicitons les isomorphismes annoncés dans Letzter (1992) et Perkins (1991). Notons que dans le cas où k est un corps algébriquement clos de caractéristique nulle, la classe des algèbres d'opérateurs differentiels (X(b)) est très importante puisque toute k-algèbre Morita équivalente à A 1 est isomorphe à un (X(b)); see Kouakou (2003).…”