Gorenstein rings as specializations of unique factorization domains

Abstract

It is known that a unique factorization domain which is a CohenMacaulay factor ring of a regular local ring is Gorenstein ((221, see, however, [23]). In this paper we show that conversely any Gorenstein ring R which is a factor ring of a regular local ring and a complete intersection locally in codimension one can be realized as a specialization of a unique factorization domain S which is Cohen-Macaulay and an epimorphic image of a regular local ring (Proposition 1). S is obtained as a local blowing-up ring of a generic link of R (for the definitions we refer to the first section of this paper). As a first application we prove that for certain singularities the highest exterior power and any symmetric power of the module of differentials are never Cohen-Macaulay (Corollary 1). In a second corollary we construct examples of “bad” unique factorization domains S such that each S is a local Cohen-Macaulay ring and the singular locus of S has “big” codimension but nevertheless the completion of S is no longer factorial. This contrasts with a result of H. Flenner [7, 1.51 saying essentially that for quasihomogeneous singularities which fulfill Serre’s conditions (S,) and (R2) factoriality is preserved under completion. Now let R be the reduced local ring of a quasihomogeneous Gorenstein singularity over a perfect field such that R is a complete intersection locally in codimension two and the degrees of the homogeneous free resolution of R are “big enough.” By S we denote the singularity constructed in Proposition 1. In this situation we can use Flenner’s result to conclude that also the completion S of S is factorial and hence Z? admits a formal deformation to the complete unique factorization domain S. This implies that rigid Gorenstein singularities satisfying the above assumptions are factorial (Proposition 2). 129 0021-8693/84 $3.00