Symplectic bundles over affine surfaces
Let A be a commutative ring. A skew-symmetric space over A is a pair (P,s) where P is a finitely generated projective A-module and \(s:\quad P\times P\to A\) a skew-symmetric bilinear form which induces an isomorphism \(s_*: P\overset \sim \rightarrow P^*=Hom_ A(P,A).\) Let L(A) be the Grothendieck group of skew-symmetric spaces, which respect to the orthogonal sum. The Witt group \(W^{-1}(A)\) is the quotient of L(A) by the subgroup generated by the hyperbolic spaces. Let now A be a real affine algebra of Krull dimension 2 and \(V=Spec A\). In his paper in Algebraic K-theory, number theory, geometry and analysis, Proc. Int. Conf., Bielefeld 1982, Lect. Notes Math. 1046, 261-328 (1984; Zbl 0531.10024), \textit{W. Pardon} relates \(W^{-1}(A)\) to the group \(A_ 0(V)\) of zero-cycles of V modulo rational equivalence. He proves that if V is smooth and if Pic V is trivial, \(W^{-1}(A)\otimes {\mathbb{Z}}/2\simeq A_ 0(V)\otimes {\mathbb{Z}}/2\). - In this paper we prove that for any real affine surface \(V=Spec A\), not necessarily smooth, \(W^{-1}(A)\otimes {\mathbb{Z}}/2\simeq SK_ 0(A)/tr K_ 0(A)\), tr\(\tilde K_ 0(A)\) denoting the subgroup of \(SK_ 0(A)\) generated by all elements of the form \(P\oplus P^*\). If Pic A is trivial, tr\(\tilde K_ 0(A)=2SK_ 0(A)\) and if V is smooth, \(SK_ 0(A)=A_ 0(V)\) hence our result extends Pardon's theorem. The description of \(W^{-1}(A)\) in terms of linear data raises the following natural question: for a projective module P over a ring A, on what conditions is the map det: Aut\(P\to A^*\) surjective ? We prove this to be the case if, for instance, P is a rank d projective A-module where A is an affine algebra of dimension d over an algebraically closed field of characteristic 0.
Grothendieck group of skew-symmetric spaces
scientific article; zbMATH DE number 3964005