On descent in dimension two and non-split Gorenstein modules

Abstract

In this paper we present a ring with a Gorenstein module but no dualizing module. The context of this example will be illustrated in the historical overview presented below. In [21] Sharp furnished the definition of a Gorenstein module G of rank t over a local, finite-dimensional ring A: G is finitely generated and for p~Spec A the Bass number, ,LL~(#z, G), equals t if i= htA(fi) and equals zero otherwise. Sharp went further to show that any ring admitting a Gorenstein module must be Cohen-Macaulay, and also must have Gorenstein formal fibres [21]. We observe that a Gorenstein module G of rank one over an n-dimensional local ring (A, m, k) induces a natural equivalence of functors Ext;( -, G)=Hom,( -, E(k)) on modules of finite length. For this reason Fossum, Foxby, Griffith, and Reiten assigned to Gorenstein modules of rank one the name dualizing module [S]. In the same paper, it was obtained that any Gorenstein A-module is a direct sum of copies of the Gorenstein A-module of minimal rank (of which, up to isomorphism, there is only one) [S]. It is clear that if a ring has a dualizing module, then any of its Gorenstein modules are direct sums of the dualizing module. Foxby [S], Reiten [19], and Sharp [21] showed that a local, CohenMacaulay ring A admits a dualizing complex if and only if A has a dualizing module. On the other hand, Grothendieck, who initiated the notion of a dualizing complex, provided the following result: