Geodesic
An invariant of quadratic forms over schemes
Documenta mathematica, Tome 1 (1996), pp. 449-478
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A ring homomorphism e0:W(X)→EX from the Witt ring of a scheme X into a proper subquotient EX of the Grothendieck ring K0​(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 e0 on the Witt class of a bundle of an endomorphisms E of an indecomposable component V0​ of the Swan sheaf U with the trace of a product as a bilinear form θ is outside of the image of composition W(F)→W(X)→E(X). Therefore the Witt class of (E,θ) is not extended. 
@article{10_4171_dm_19,
     author = {Marek Szyjewski},
     title = {An invariant of quadratic forms over schemes},
     journal = {Documenta mathematica},
     pages = {449--478},
     year = {1996},
     volume = {1},
     doi = {10.4171/dm/19},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/19/}
}
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Marek Szyjewski. An invariant of quadratic forms over schemes. Documenta mathematica, Tome 1 (1996), pp. 449-478. doi: 10.4171/dm/19

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