Twisted Rings of Differential Operators over Projective Rational Curves Original Research Article

Let image be a singular, rational, projective curve over an algebraically closed field k of characteristic zero. M. P. Holland and J. T. Stafford (J. Algebra147, 1992, 176-244) described the twisted ring of differential operators imageimage(image) for image an invertible sheaf over image in the case when the normalization map π: image1 → K is injective. In this paper we consider rational curves with no restrictions on the normalization map pi. If image is an open affine subset of image, it is well-known that image(image) has a unique, minimal non-zero ideal J(image). And so the ring structure of image(image) is determined by the factor F(image) = image(image)/J(image) as described by K. A. Brown (Math. Z. 206, 1991, 424-442). If we let image0 be an open affine subset of image containing all the singular points, we have the following: THEOREM A. If image has sufficiently high degree, then[formula], where Jimage(image) is the unique minimal non-zero ideal of imageimage(image) and F(image0) = imageimage0)/ J(image0) is as described above. Moreover, an analog of Beilinson and Bernstein′s equivalence of categories holds, namely: THEOREM B. If image has sufficiently high degree, then: (1) The category of quasi-coherent sheaves of imageimage-modules is equivalent to the category of finitely generated left imageimage(image)-modules. (2) The right imageimage(image)-module imageimage(image0) circled plus imageimage(image1) is faithfully flat, where image = image0 union or logical sum image1is an open affine cover. The equivalence of categories is independent of image in the following sense: THEOREM C. If image and image have sufficiently high degree, then imageimage(image) and imageimage(image) are Morita equivalent.