Hermitian forms in additive categories: Finiteness results
In a ring E with involution \({}^*\) the set of units fixed by the involution is denoted by \(S(^*, E^{\times})\). Two elements \(u,v\in S(^*, E^{\times})\) are similar if there is some \(w\in E^{\times}\) such that \(u=w^*vw\). The set of similarity classes of \(S(^*, E^{\times})\) is denoted by \(H(^*, E^{\times})\). These similarity classes are related to isometry classes of certain Hermitian forms [cf. the authors, J. Reine Angew. Math. 375/376, 167-183 (1987; Zbl 0607.57013), section 5]. If E is Q-finite, i.e., the torsion group of E is finite and \(E\otimes_{{\mathbb{Z}}} {\mathbb{Q}}\) is finite dimensional over \({\mathbb{Q}}\), then \(E_ p\) denotes the p-adic completion of E at the rational prime p. There is a canonical localization map \(H(^*, E^{\times})\to \prod_{p}H(^*, E_ p^{\times})\). It is shown that the fibres of this map are finite. The paper also contains a number of related finiteness results as well as applications to forms with group actions, to systems of forms and to knot theory.
scientific article; zbMATH DE number 4128905