Excellent special orthogonal groups (Q5950388)

Let \(F\) be a field of characteristic \(\neq 2\) and let \(\varphi\) be an anisotropic quadratic form over \(F\). The higher kernel forms \(\varphi_i\) are defined recursively as follows. One puts \(\varphi_0=\varphi\), \(F_0=F\), and \(F_1=F(\varphi)\) the function field of \(\varphi\) over \(F\). For \(i\geq 1\), one defines \(\varphi_i\) as the anisotropic part of \(\varphi_{i-1}\) over \(F_i=F_{i-1}(\varphi_{i-1})\). The smallest \(h\) with \(\dim\varphi_h\leq 1\) is called the height of \(\varphi\). Following \textit{M. Knebusch} [Proc. Lond. Math. Soc. (3) 33, 65-93 (1976; Zbl 0351.15016); 34, 1-31 (1977; Zbl 0359.15013)], an anisotropic form \(\varphi\) is called excellent if all higher kernel forms are defined over \(F\), i.e. for all field extensions \(E/F\) there exists a form \(\psi\) over \(F\) such that the anisotropic part of \(\varphi_E\) is isometric to \(\psi_{E}\). Knebusch has shown that an anisotropic form \(\psi=\psi_0\) is excellent iff there exists a sequence \(\psi_0, \psi_1,\cdots,\psi_h\), \(\dim\psi_h\leq 1\) such that \(\dim\psi_i>\dim\psi_{i+1}\) and \(\psi_i\perp -\psi_{i+1}\) is similar to an anisotropic Pfister form. NEWLINENEWLINENEWLINEIn the present paper, a form \(\varphi\) is called quasi-excellent if, up to similarity over \(E\), all higher kernel forms are defined over \(F\). It is not difficult to show that for odd-dimensional forms, the notions of excellence and quasi-excellence coincide. This is generally no longer true in even dimensions. A sequence of anisotropic forms \(\varphi=\varphi_0,\varphi_1 ,\cdots,\varphi_h\) is called quasi-excellent if the anisotropic part of \(\varphi\) over \(F(\varphi_0,\cdots ,\varphi_{i-1})\) is similar to \(\varphi_i\) and \(\dim\varphi_h=0\). In this case, \(\varphi_{h-1}\) is similar to a Pfister form whose degree is called the degree of that sequence, and furthermore, the form \(\varphi_0\) is quasi-excellent. NEWLINENEWLINENEWLINEThe aim of the paper is to provide a classification of all quasi-excellent sequences based on the claim that all such sequences can be constructed (in the reversed order of increasing dimension) by adding to a sequence ending in \(\psi\) a form \(\psi'\) such that \(\psi'\) is a Pfister neighbor with complementary form \(\psi\), starting from three different types of short sequences having \(2\), \(3\) and \(4\) terms, respectively. It is shown that quasi-excellent sequences ending in such a short sequence of the first type are always obtained by this method, whereas for the two other types this is proved if the degree is \(1\) and \(2\) (the latter by unpublished results by Rost). It is furthermore shown that this holds in any degree provided the characteristic of \(F\) is zero. This follows by applying results by \textit{V. Voevodsky} [\(K\)-theory Preprint Archives No. 502 (2000)] and \textit{D. Orlov, A. Vishik} and \textit{V. Voevodsky} [\(K\)-theory Preprint Archives No. 454 (2000)] concerning the Milnor conjecture. NEWLINENEWLINENEWLINEThe interest in this problem is motivated by the following. In [J. Algebra 200, 334-346 (1998; Zbl 0910.20027)], the second author and \textit{U. Rehmann} coined the notion of excellence for semisimple algebraic groups and studied it for special linear and special orthogonal groups. It was shown that \(SO(\varphi)\) is excellent iff (with the above definition) \(\varphi\) is quasi-excellent. Thus, the above results provide criteria for the excellence of special orthogonal groups. NEWLINENEWLINENEWLINEThe results of the paper are based on a draft by Oleg Izhboldin brought into final shape by the second author after Izhboldin's tragic and untimely death.