ALGEBRAIZATIONS WITH MINIMAL CLASS GROUP

Abstract

Question. Let R be a complete normal local ring with coefficient field C . Does there exist a local ring Al essentially of finite type over C , such that the class group C1 (A) of A is generated by the canonical module w~ of A and its completion A E R? In general one knows that Cl(A) + Cl(R) is injective (see [I]) and the question of how small one can make Cl(A) arises. Srinivas has constructed UFD's (i.e., Cl(A) = 0) with arbitrary rational double point singularities in his study of the K-theory of these singularities (see [15]). KollAr conjectured that any isolated hypersurface singularity would have an UFD globalization and some partial results were obtained by Buium (see [2]). In [12] the first author and Srinivas settled the above question in the affirmative for isolated complete intersection singularities. Recently, Heitmann (see [6]) has constructed, for any complete local ring R over C of depth at least two, UFD's with completion R. But these rings are not geometric in general and they do not have dualizing modules. Indeed, a theorem of Murthy asserts that a geometric Cohen-Macaulay UFD is Gorenstein (see [ l l ] ) . So for non-Gorenstein R it seems more natural to look for geometric Als with class group generated by the canonical module. In this note we will prove that the above question has an affirmative answer in the case of normal surface singularities: