Generic splitting for special groups
The theory of special groups has been developed by \textit{M. A. Dickmann} and \textit{F. Miraglia} [Special groups. Boolean-theoretic methods in the theory of quadratic forms. Mem. Am. Math. Soc. 689 (2000; Zbl 1052.11027)]. Categorically, it is equivalent to the theory of abstract Witt rings introduced by Marshall. The purpose of the present paper is to develop a theory of generic splitting for special groups that mimics the classical theory of generic splitting of quadratic forms over fields developed by \textit{M. Knebusch} in two seminal papers [Proc. Lond. Math. Soc., III. Ser. 33, 65--93 (1976; Zbl 0351.15016), ibid. 34, 1--31 (1977; Zbl 0359.15013)]. The author defines an analogue of the notion of generic zero (or partial splitting) field over a field \(k\) of a quadratic form \(q\) over \(k\) based on model-theoretic considerations. If \(q\) is a form of dimension at least \(2\) defined over a special group \(G\), then a generic partial splitting (GPS) of \(q\) over \(G\) is a morphism of special groups \(\lambda : G\to K\) with \(\lambda (q)\) isotropic that can be characterized by the property that to any morphism of special groups \(f:G\to L\) with \(f(q)\) isotropic and such that \(L\) is \(|K|^+\)-saturated, there exists a morphism of special groups \(\theta: K\to L\) with \(f=\theta\circ\lambda\). Using this definition, it is also possible to construct generic splitting towers of special groups for forms over special groups, and to introduce higher Witt indices and the height of forms over special groups with properties analogous to those valid in the case of generic splitting of quadratic forms over fields. For reduced special groups, the author also obtains a series of results that are well known in the case of quadratic forms over fields. Let \(\varphi\) and \(\psi\) be forms over a reduced special group \(G\) such that \(\dim\varphi\geq 2\), \(1\) is represented by \(\varphi\), and \(\varphi\) and \(\psi\) anisotropic. Let \(\lambda:G\to G_\varphi\) be a GPS for \(\varphi\). It is shown that if \(\lambda (\psi)\) is hyperbolic, then \(a\varphi\) is a subform of \(\psi\) for any \(a\) represented by \(\psi\) (subform theorem), that \(\lambda (\varphi)\) is hyperbolic iff \(\varphi\) is a Pfister form (characterization of anisotropic forms of height \(1\)), and, combining these two results, that if \(\varphi\) is Pfister and \(\lambda (\psi)\) is hyperbolic, then \(\psi\equiv\rho\otimes\varphi\) for some form \(\rho\) over \(G\), showing that the kernel of the restriction map \(W(G)\to W(G_\varphi)\) is given by \(\varphi W(G)\). The paper finishes with a study of certain functorial properties of GPS in the case of special groups that are not necessarily reduced.
scientific article; zbMATH DE number 5353589
scientific article