K-theory of quadric hypersurfaces
The main object of this paper is to extend Quillen's calculation of the higher K-theory of projective spaces to smooth quadratic hypersurfaces [\textit{D. Quillen}, in Algebraic K-theory I, Proc. Conf. Battelle Inst. 1972, Lect. Notes Math. 341, 85-147 (1973; Zbl 0292.18004)], by proving the following theorem: Let \(X\subset P_ R^{d+1}\) be a quadratic hypersurface, defined by a quadratic form q over a commutative ring R. Assume that X is smooth of relative dimension \(d\) over R. Then \(K_ i(X)\approx K_ i(R)^ d\oplus K_ i(C_ 0(q))\) where \(C_ 0(q)\) is the even part of the Clifford algebra of q. In fact a more general version applicable to \({\mathbb{H}}\)-bundles and to quadratic forms on projective (rather than free) modules is given. The proof is based on that of Quillen and therefore gives an affirmative answer to the first problem posed by the author in his paper appearing in the proceedings of the John Moore Conference, {\S} 13. The second problem, to determine the Chow ring of X, is still open. The theorem is used to get information on the K-theory of affine quadratic hypersurfaces by using the localization sequence, which in particular provides a proof of the main conjecture in the author's cited paper, {\S} 7. From this the following theorem is deduced: For \(\Lambda ={\mathbb{R}}\), \({\mathbb{C}}\) or \({\mathbb{H}}\), \(K_ 0\oplus_{{\mathbb{R}}}R_ n\approx K^ 0_{\Lambda}(S^ n)\) where \(R_ n={\mathbb{R}}[X_ 0,...,X_ n]/(\sum x^ 2_ i-1)\quad is\) the ring of polynomial functions on \(S^ n.\) The case \(\Lambda ={\mathbb{C}}\) is due to \textit{R. Fossum} [Invent. Math. 8, 222-225 (1969; Zbl 0172.485)]. This theorem was the original aim of most of the work in the author's paper cited above, as well as the present paper. Another application of the first theorem is to compute the K-theory of coherent sheaves on the affine cone over the hypersurface given by q.
scientific article; zbMATH DE number 3968749