Twisted Rings of Differential Operators over Projective Rational Curves

Abstract

Let X be a singular, rational, projective curve over an algebraically closed field k of characteristic zero. M. P. Holland and J. T. Stafford ( J. Algebra 147, 1992, 176-244) described the twisted ring of differential operators D L ( X ) for L an invertible sheaf over X in the case when the normalization map π: P 1 → K is injective. In this paper we consider rational curves with no restrictions on the normalization map pi. If U is an open affine subset of X , it is well-known that D ( U ) has a unique, minimal non-zero ideal J( U ). And so the ring structure of D ( U ) is determined by the factor F( U ) = D ( U )/ J( U ) as described by K. A. Brown ( Math. Z. 206, 1991, 424-442). If we let U 0 be an open affine subset of X containing all the singular points, we have the following: THEOREM A. If L has sufficiently high degree, then[formula], where J L ( X ) is the unique minimal non-zero ideal of D L ( X ) and F( U 0) = D U 0)/ J( U 0) is as described above. Moreover, an analog of Beilinson and Bernstein′s equivalence of categories holds, namely: THEOREM B. If L has sufficiently high degree, then: (1) The category of quasi-coherent sheaves of D X - modules is equivalent to the category of finitely generated left D L ( X )- modules. (2) The right D L ( X )- module D L ( U 0) ⊕ D L ( U 1) is faithfully flat, where X = U 0 ∪ U 1 is an open affine cover. The equivalence of categories is independent of L in the following sense: THEOREM C. If L and F have sufficiently high degree, then D L ( X ) and D F ( X ) are Morita equivalent.