An invariant of quadratic forms over schemes

Abstract

A ring homomorphism $e^0:W(X)\to EX$ from the Witt ring of a scheme $X$ into a proper subquotient $EX$ of the Grothendieck ring $K_0(X)$ is a natural generalization of the dimension index for a Witt ring of a field. In the case of an even dimensional projective quadric $X$, the value of $e^0$ on the Witt class of a bundle of an endomorphisms $\mathcal{E}$ of an indecomposable component ${\mathcal{V}}_0$ of the Swan sheaf $\mathcal{U}$ with the trace of a product as a bilinear form $\theta$ is outside of the image of composition $W(F)\to W(X)\to E(X)$. Therefore the Witt class of $(\mathcal{E},\theta )$ is not extended.