Witt groups of projective line bundles (Q5949802)

Consider the structure morphism \(f:Y\to X\), where \(X\) is a noetherian scheme over which \(2\) is invertible, and \(Y=\mathbb{P}({\mathcal S})\) is the projective Grothendieck line bundle corresponding to a vector bundle \({\mathcal S}\) of rank \(2\) over \(X\). The aim of this paper is to show that the induced morphism of Witt rings \(f^*:W(X)\to W(Y)\) fits into an exact sequence \(W(X)\to W(Y)\to M_{\top}(X)\) with \(M_{\top}(X)\) being a Witt group of formations over \(X\) similar to the one defined by \textit{A. Ranicki} [Exact Sequences in the Algebraic Theory of Surgery, Math. Notes 26, Princeton University Press (1981; Zbl 0471.57012)] in the affine case, but equipped with a duality functor which might differ from the usual one. The author points out that in a private communication to him, C.~Walter announced results on higher Witt groups of projective space bundles over \(X\) which subsume the results in the present paper. The author mentions that in the case where \({\mathcal S}\) has a quotient bundle of rank \(1\) so that there is a section \(X\to Y\), then \(W(X)\to W(Y)\) is a monomorphism. This holds, for example, if \(Y=\mathbb{P}_X^1\) is the trivial projective line bundle. According to Walter and under the same hypothesis on \({\mathcal S}\), the map \(W(Y)\to M_{\top}(X)\) is an epimorphism, so that in this special case the above exact sequence extends to a short exact sequence.