An invariant of quadratic forms over schemes (Q1355997)

Let \(X\) be a scheme. The dualization functor induces a ring homomorphism \({}^\wedge: K_0X \to K_0X\). Let \(EX= \text{Ker} (1-{}^\wedge)/ \text{Im} (1+{}^\wedge)\). The author defines a ring homomorphism \(e_0:WX\to EX\) which is a natural generalization of the dimension index in the theory of quadratic forms. He computes \(EX\) and \(e_0\) for regular curves \(X\). In this case \(e_0\) is surjective, but if \(EX\neq \mathbb{Z}/2\mathbb{Z}\) there are bilinear spaces on \(X\) which are not induced from the ground field \(F\). Methods from algebraic geometry can be used to compute \(EX\) for many quasiprojective schemes \(X\). In particular this is done very explicitly for projective quadrics of maximal index \(m\) for any dimension \(2m\) resp. \(2m+1\). Here the Swan bundle and Clifford algebras play a significant role. For \(\dim X= 2m>2\) we have \(EX= (\mathbb{Z}/2\mathbb{Z})^2\), and \(e_0\) is surjective. Non-induced bilinear bundles can be computed explicitly. Furthermore there are bilinear bundles which are hyperbolic everywhere locally, but not globally.