Generic splitting towers and generic splitting preparation of quadratic forms (Q2704204)

The first goal of this paper is to develop a generic splitting theory for regular quadratic forms over fields of arbitrary characteristic. This extends previous work of \textit{M. Knebusch} [Proc. Lond. Math. Soc. (3) 33, 65-93 (1976; Zbl 0351.15016); ibid. 34, 1-31 (1977; Zbl 0359.15013)] over fields of characteristic different from 2, although the unified approach in the present paper is somewhat different.NEWLINENEWLINENEWLINEThe authors develop the notion of a generic splitting tower of a form \(q\) over an arbitrary field \(k\) and explain how such a tower, together with the sequence of associated higher kernel forms, controls the splitting of \(q \otimes L\) into a sum of hyperbolic planes and an anisotropic form, for any field extension \(L\) of \(k\). It is then proved that for a suitable generic splitting tower of \(q\) every associated higher kernel form of \(q\) is definable over some finitely generated purely transcendental extension of \(k\).NEWLINENEWLINENEWLINEThis result leads to the second main topic of the paper, the notions of generic splitting preparations and generic splitting decompositions. A generic splitting decomposition of a form \(q\) over \(k\) consists of a purely transcendental field extension \(K/k\) and an orthogonal decomposition of \(q\otimes K \cong \eta_0 \perp \eta_1 \perp \dots \perp \eta_h \perp \varphi_h\) satisfying certain conditions. In particular, \(\text{dim} \varphi_h\leq 1\), all \(\eta_i\) have even dimension, and \(\eta_0\) is the hyperbolic part of \(q\otimes K\). It is shown that the generic splitting decomposition controls the splitting behavior of \(q\otimes L\), for any field extension \(L\) of \(k\), in a sense made explicit using the notion of quadratic places [see \textit{M. Knebusch}, J. Reine Angew. Math. 517, 117-130 (1999; Zbl 0978.11013)]). The use of quadratic places necessitates the restriction to fields of characteristic different from 2 for the final part of the paper.NEWLINENEWLINEFor the entire collection see [Zbl 0956.00036].