Gorenstein Witt Rings

Abstract

Throughout R is a noetherian Witt ring. The basic example is the Witt ring WF of a field F of characteristic not 2 and finite. We study the structure of (noetherian) Witt rings which are also Gorenstein rings (i.e., have a finite injective resolution). The underlying motivation is the elementary type conjecture. The Gorenstein Witt rings of elementary type are group ring extensions of Witt rings of local type. We thus wish to compare the two classes of Witt rings: Gorenstein and group ring over local type. We show the two classes enjoy many of the same properties and are, in several cases, equal. However we cannot decide if the two classes are always equal.In the first section we consider formally real Witt rings R (equivalently, dim R = 1). Here the total quotient ring of R is R-injective if and only if R is reduced. Further, R is Gorenstein if and only if R is a group ring over Z. This result appears to be somewhat deep.