An invariant of quadratic forms over schemes

  • Marek Szyjewski

    Instytut Matematyki Uniwersytet Slaski PL 40007 Katowice, ul. Bankowa 14, Poland
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A ring homomorphism e0:  W(X)EXe^0:\; W(X)\rightarrow EX from the Witt ring of a scheme XX into a proper subquotient EXEX of the Grothendieck ring K0(X)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 XX, the value of e0e^0 on the Witt class of a bundle of an endomorphisms E\mathcal{ E} of an indecomposable component V0\mathcal{ V}_0 of the Swan sheaf U\mathcal{ U} with the trace of a product as a bilinear form θ\theta is outside of the image of composition W(F)W(X)E(X)W(F)\rightarrow W(X)\rightarrow E(X). Therefore the Witt class of (E,θ)(\mathcal{ E},\theta) is not extended.

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Marek Szyjewski, An invariant of quadratic forms over schemes. Doc. Math. 1 (1996), pp. 449–478

DOI 10.4171/DM/19