A ring homomorphism e^0: W(X)--> EX from the Witt ring of a scheme X into
proper subquotient EX of the Grothendieck ring K_0(X) is a natural
generalisation of dimension index for Witt ring of a field. In the case of
even dimensional projective quadric X over a field F the value of e^0 on the
Witt class of bundle of endomorphisms E of an indecomposable component V_0 of
the Swan sheaf U with trace of product as a bilinear form \theta is outside
of the image of composition W(F)--> W(X)--> E(X). Therefore the Witt class of
(E,\theta) is not extended from F.
This paper has appeared as paper 19 in Documenta
Mathematica, volume 1.
This paper has appeared as paper 19 in Documenta Mathematica, volume 1.