Vanishing and nilpotence of locally trivial symmetric spaces over regular schemes
The author uses his theory of triangular Witt groups [\textit{P. Balmer}, \(K\)-Theory 19, 311--363 (2000; Zbl 0953.18003), Math. Z. 236, 351--382 (2001; Zbl 1004.18010)] to reprove and improve two classical results on Witt rings of regular schemes, in which 2 is invertible. In the first result \(R\) is a noetherian semi-local ring containing \(\frac{1}{2}\). Let \(U\) be the punctured spectrum obtained from \(\operatorname{spec} R\) by removing the maximal ideals of maximal height. Then the natural homomorphism \(W(R) \rightarrow W(U)\) is injective. As a consequence, this result implies that the map \(W(R) \rightarrow W(Q)\) from the Witt ring of \(R\) to the Witt ring of its quotient field \(Q\) is injective, provided \(R\) is a noetherian, regular, semi-local ring of Krull dimension \(\leq 4\). To describe the second main result, let \(X\) be a regular integral scheme of finite Krull dimension with function field \(Q\). The kernel of the natural map \(W(X) \rightarrow W(Q)\) is known to be nilpotent by results of \textit{T. C. Craven}, \textit{A. Rosenberg} and \textit{R. Ware} [Proc. Am. Math. Soc. 51, 25--30 (1975; Zbl 0313.13025)] and \textit{M. Knebusch} [in: Proc. Conf. Quadratic Forms, Kingston 1976, Queen's Pap. pure appl. Math. 46, 103--283 (1977; Zbl 0408.15019)]. The author shows that there exists an integer \(N\), depending only on \([\frac{\dim X}{4}]\), so that the \(N\)th power of the kernel is zero. For \(0 \leq \dim X \leq 7\) the integer \(N\) can be taken to be \(N = [\frac{\dim X}{4}] + 1\), and the author gives examples of arbitrary dimension, which show that in general \(N = [\frac{\dim X}{4}] + 1\) is the best possible exponent.
scientific article; zbMATH DE number 1990087