On the isomorphism problem for rings of differential operators over affine curves

摘要

In the group Aut(k)(A(1)(k)) of k-automorphisms of A(1) :=A(1)(k) (the first Weyl algebra on a field k of any characteristic p >= 0), we solve the following problem: Find sigma is an element of Aut(k)(A(1)), such thatsigma(A(1) + A(1)partial derivative(-1)b) =A(1) + A(1)partial derivative(-1)t(n)where 0 not equal b is an element of k[t] and n = deg(b). This problem is a particular case of the general problem of Stafford (1987) on isomorphisms between two k-algehras D and D' 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 D(X(h)). Due to the resolution of the problem above, we find the condition to have an isomorphisin between two such algebras of differential operators. 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 D(X(b)) is a very important one, since any k-algebra B Morita equivalent to A I is isomorphic to some D(X(h)) in Kouakou (2003).