An Invariant of Quadratic Forms over Schemes

A ring homomorphism $e^0:\; W(X)\rightarrow EX$ 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 $W(F)\rightarrow W(X)\rightarrow E(X)$. Therefore the Witt class of $(\mathcal{ E},\theta)$ is not extended.

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