Grothendieck and Witt rings of hermitian forms over Dedekind rings

Abstract

The prime ideal theory of the Grothendieck and Witt ring of non-degenerat e hermitian forms over a Dedekind ring with involution is studied. The relationship of these rings to those defined over the quotient field of the Dedekind ring is also investigated. The main goal of this paper is to extend the structure theory for Witt rings over fields of Poster [18] and Harrison-Leicht-Lorenz ([10], [16]) to the Grothendieck ring K(C, J) and the Witt ring W(C, J) of a Dedkind ring C with an involution J. Since the case J = identity is allowed, the Grothendieck and Witt rings of [12] are included. We shall see that the main theorems of Pfister and Harrison-LeichtLorenz remain true for W(C, J) and that if J is the identity they are also true for K(C, J). However, for K(C, J) with J ^ identity there is some deviation: there may be p-torsion for primes p 2 and there may be nilpotent elements which are not torsion (Example 1.3). This fact has been overlooked in [13]. In §1 we extend some elementary results of [12, §11, §13] to the case J ^ identity. We conjecture that they are well known to the specialists but we did not find an appropriate reference in the literature. We show that the canonical map from W(C, J) to the Witt ring W(L, J) of the quotient field L of C is infective and give some information about the kernel A(C, J) of the map K(C, J) —> K(L, J). Since the exact determination of A(C, J) is not needed for our structure theory we delay this matter to §4, where such a deter­ mination is given along the same lines as in [12, §11.2], We then show that W(C, J) is the intersection of certain subrings W(C„ J) of W(L, J) which are Witt rings for abelian groups of exponent 2 in the sense of [14, Def. 3.12] and we describe the image K'(Cf J) of K{C, J) in K(L, J) in an analogous way. We are thus led to study subrings T of an abstract Witt ring R for an arbitrary abelian g-group [14, Def. 3.12]. If T is the inter­ section of a family {Ta} of subrings of R which are also Witt rings for some abelian g-groups, the entire prime ideal theory of R remains true for T. In § 3 we show that if T is either K(C, J) or W(C, J) then the group of units of T is generated by 1 + Nil T and the rank one spaces