Specialization of quadratic and symmetric bilinear forms. Transl. by Thomas Unger
In a series of seminal papers in the 1970s [Acta Arith. 24, 279--299 (1973; Zbl 0287.15010)], [Proc. Lond. Math. Soc. (3) 33, 65--93 (1976; Zbl 0351.15016)], Proc. Lond. Math. Soc., III. Ser. 34, 1--31 (1977; Zbl 0359.15013)], \textit{M. Knebusch} initiated a systematic theory of specialization of quadratic forms and their generic splitting theory, in particular the study of quadratic forms over function fields of quadrics, a theory that has blossomed ever since, in particular from the 1990s on in the work of Izhboldin, Karpenko, Merkurjev, Laghribi, Vishik and the reviewer to name but a few, and that has yielded some spectacular breakthroughs, partly based on the incorporation of algebro-geometric techniques introduced by Rost, Voevodsky and others that also lead to Voevodsky's proof of the Milnor conjecture. The present book represents the state of the art in the theory of specialization of quadratic forms, containing a substantial amount of previously unpublished new results. It is a welcome addition to the literature as this is the first monograph in which this original and important approach to the algebraic theory of quadratic forms has been given such a comprehensive treatment. In the sequel, we give an overview of the contents of the book. In Chapter 1, the author introduces the basic idea of specialization: Let \(\lambda\,:\,K\to L\cup\infty\) be a place with valuation ring \({\mathfrak o}\subset K\) and maximal ideal \({\mathfrak m}\). Let \(\varphi\) be a symmetric bilinear form over \(K\). We say that \(\varphi\) has good reduction if \(\varphi\) is isometric to \((a_{ij})\) with \(a_{ij}\in {\mathfrak o}\) and \(\det (a_{ij})\in {\mathfrak o}^*\). This gives rise to a well-defined (up to isometry) form \(\lambda_*(\varphi)=(\lambda (a_{ij}))\) provided \(\text{char}(L)\neq 2\). If \(\text{char}(L)=2\), then at least the anisotropic part of \(\lambda_*(\varphi)\) is well defined up to isometry. Thus, in any case, \(\lambda\) induces an additive map between the Witt rings \(W(K)\to W(L)\). If \(\text{char}(L)\neq 2\), then quadratic forms and bilinear forms can be identified and so this also gives rise to a specialization theory of quadratic forms. This quickly leads to the theory of generic splitting of quadratic forms in characteristic not \(2\) (Ch. 1.4) where one studies the isotropy behaviour of quadratic forms under field extensions, for example the splitting pattern of a quadratic form, i.e., the set of integers that occur as Witt indices of a given quadratic form when passing to all possible field extensions. The main problem arises when considering the behaviour of \textit{quadratic} forms under places \(\lambda\,:\,K\to L\cup\infty\) when \(\text{char}(L)= 2\). After a short introduction to quadratic modules (especially over valuation rings) in Ch. 1.5--6, the author refines the notion of good reduction in Ch.~1.7 by defining a quadratic space \(E\) over \(K\) to have good reduction with respect to \({\mathfrak o}\) if \(E\cong K\otimes_{\mathfrak o} M\) for some nondegenerate (in a suitably defined sense) quadratic \({\mathfrak o}\)-module \(M\). This notion is extended to a much larger class of forms that are called \textit{obedient} with respect to \(\lambda\) and which allow an orthogonal decomposition into certain scalar multiples of forms with good reduction. Using this decomposition, it is now possible to define a so-called weak specialization \(\lambda_W(E)\) of an obedient form \(E\) which is a quadratic space over \(L\) uniquely determined up to Witt equivalence (i.e. whose anisotropic part is uniquely determined up to isometry). This yields nothing new if \(\text{char}(L)\neq 2\), but it allows to give a meaningful notion of specialization for quadratic forms under places \(\lambda\,:\,K\to L\cup\infty\) when \(\text{char}(L)= 2\). Using these notions, the author finishes the chapter by studying good reduction of spaces under orthogonal sum and for products of a bilinear space times a quadratic space with suitable regularity resp. obedience properties (Ch. 1.8). Chapter 2 is devoted to generic splitting of quadratic forms over fields of characteristic \(2\). In this situation, a quadratic form \(\varphi\) is nondegenerate in the author's sense if its quasilinear part \(\text{QL}(\varphi)\) is anisotropic. The function field \(K(\varphi)\) of a quadratic form is introduced and the notion of a generic separable zero field of a form \(\varphi\). It is shown that if \(\varphi\neq\text{QL}(\varphi)\), then \(K(\varphi)\) is a generic separable zero field of \(\varphi\) (Ch. 2.1--2). The author points out certain limitations of the specialization theory so far, especially when treating places \(\lambda\,:\,K\to L\cup\infty\) with \(\text{char}(K)= 0\) and \(\text{char}(L)= 2\). For example, it is not clear how to ``lift'' forms \(\psi\) over \(L\) with \(\text{QL}(\psi)\neq 0\) to forms over \(K\). To this end, the author introduces the notion of fair reduction of a quadratic module \(E\) over \(K\), meaning that there exists a quadratic module \(M\) over \({\mathfrak o}\) with \(M/{\mathfrak m}M\) nondegenerate such that \(E\cong K\otimes_{\mathfrak o}M\). (Ch. 2.3). One obtains now a unified generic splitting theory in which one studies the isotropy behaviour of nondegenerate forms over those field extension over which the quasilinear part stays anisotropic, leading to the notions of splitting patterns, height, splitting towers but now in a more general context than previously (Ch. 2.4--2.6). Chapter 3 is devoted to a range of applications and results, some of them new or lesser known characteristic \(2\) versions of results that are well known in characteristic not \(2\), including a study of forms of height \(1\), the Cassels-Pfister subform theorem, Knebusch's own norm principle, strongly multiplicative and Pfister forms and multiples of Pfister forms, Pfister neighbors and excellent forms, the notion of leading form and degree. Some open problems are also mentioned. Chapter 4 is quite short and deals with a generalization of the notion of place, namely that of a \textit{quadratic place} consisting of a triple \(\Lambda=(\lambda, H,\chi)\) where \(\lambda\,:\,K\to L\cup\infty\) is a place, \(H\) is a subgroup of the square class group of \(K\) containing that of the valuation ring \({\mathfrak o}\), and \(\chi\) is a group homomorphism from \(H\) to the square class group of \(L\) compatible with \(\lambda\) on \({\mathfrak o}^*\). This gives rise to the definitions of (stably) almost good reduction, fair reduction, stably conservative reduction with respect to \(\Lambda\). These are all rather technical yet very subtle notions that allow further broadening of the specialization and generic splitting theory of quadratic forms, in particular in the case when \(\text{char}(L)=2\).
scientific article; zbMATH DE number 5769117