An exact sequence for \(K^M_*/2\) with applications to quadratic forms
The paper under review is a natural extension of the third author's fundamental paper ``Motivic cohomology with \(\mathbb{Z}/2\)-coefficients'' [Publ. Math., Inst. Hautes Étud. Sci. 98, 59--104 (2003; Zbl 1057.14028)], in which he had presented a refined version of his spectacular proof of the famous Milnor conjecture in algebraic \(K\)-theory. Accordingly, the authors of the present paper freely refer to the methods and results developed in that foregoing work without reproducing them here. Actually, great deal of the research presented in the paper under review was conducted as early as in the spring of 1995, when all three authors were staying at Harvard University, and the final version, in its present form, was submitted a few years ago, when these three authors continued their collaboration at the IAS in Princeton. In this regard, the paper seems to have some remarkable history which probably is well-known to most leading experts in the field. As for the topics treated in the present paper, the set-up is as follows: Let \(k\) be a field of characteristic zero. For a sequence \(\underline a= (a_1,\dots,a_n)\) of invertible elements of \(k\) one has a natural homomorphism \(K^M_*(k)/2\to K^M_{*+n}(k)/2\) in Milnor's \(K\)-theory modulo elements divisible by 2, which is given by multiplication with the symbol corresponding to the sequence \(\underline a\). The main goal of the present paper is then to construct a four-term exact sequence involving the above natural homomorphism as middle term, from which some useful information about the kernel and the cokernel of it can be obtained. The proof of the existence of such an explicit exact sequence is technically rather involved and consists of two independent parts. At first, the authors define the so-called norm quadric \(Q_{\overline a}\) associated to the sequence \(\underline a\) and, using the techniques from V. Voevodsky's above-mentiond paper, derive then a four-term exact sequence relating the kernel and the cokernel of the multiplication by the symbol of \(\underline a\) with the Milnor \(K\)-groups of the closed and the generic points of the scheme \(Q_{\underline a}\), respectively. In the sequel, using further geometric arguments, it is shown that reduction to points with residue field of degree 2 yields a four-term exact sequence in Milnor's \(K\)-theory modulo 2, just as wanted. Together with the well-known classical theorem of \textit{H. Bass} and \textit{J. Tate} [in: Algebr. \(K\)-Theory II, Proc. Conf. Battelle Inst. 1972, Lect. Notes Math. 342, 349--446 (1973; Zbl 0299.12013)], this exact sequence is used to describe the generators of the kernel of the multiplication by the symbol of the sequence \(\underline a\). In the second part of the present paper, the authors give three important applications of their results established in the first part. In fact, they provide proofs of the validity of three major conjectures in the theory of quadratic forms, which demonstrates the great progress that the authors' approach has brought about. Namely, as a first corollary, an amazingly short proof of Milnor's conjecture on the structure of the Witt ring of quadratic forms over the ground field \(k\) is obtained, thereby completing foregoing partial results by J. Milnor (1969/1970), M. Rost (1986), A. Merkurjev and A. Suslin (1991) and others. The second application yields a proof of the so-called Kahn-Rost-Sujatha conjecture [\textit{B. Kahn}, \textit{M. Rost} and \textit{R. J. Sujatha}, Am. J. Math. 120, No. 4, 841--891 (1998; Zbl 0913.11018)] by establishing the following theorem: Let \(Q\) be an \(m\)-dimensional quadric over a field \(k\) of characteristic zero. Then the kernel of the natural map \(K^M_i(k)/2\to K^M_i(k(Q))/2\) is trivial for any \(i<\log_2(m+ 2)\). Finally, the third application makes the so-called \(J\)-filtration conjecture on Witt rings of quadratic forms into a well-established theorem. This concludes the previous related work by \textit{M. Knebusch} [Proc. Lond. Math. Soc., III. Ser. 33, 65--93 (1976; Zbl 0351.15016)] and by \textit{W. Scharlau} [Quadratic and Hermitean forms, Grundlehren der Mathematischen Wissenschaften, 270 (1985; Zbl 0584.10010)], who once formulated this conjecture on the filtrations of the Witt ring \(W(k)\).
applications of methods of algebraic \(K\)-theory
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