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Example text

V) is a bundle * (f E,f*v). Since Chern classes are integral and hence real. and the exterior derivative is a real operator. we can take the real part of the characteristic classes described above. ffiIZ) for p We obtain classes in A cp(E,v)rn ~ 0 which are characterized by axioms similar to those above except that i) is replaced by: i)rn If E is a line bundle and to the zero section of E. v) = (div(s). Re{2v~ }). rn». represent the Chern classes in is already real. for example if Hermitian metric on E.

Algebraic K-theory. Benjamin. New York. 1968. : 'The Picard group of a reduced G-algebra', to appear in J. Pure Applied Algebra. [FL] Fulton, W. , Springer. 1985. : Cohomologie non abelienne, Grundlehren der Math, Springer, 1971. , 1977. , Springer, 1966. : Etale Cohomology, Princeton U. Press, Princeton, 1980. : Introduction to Algebraic K-theory, Ann. of Math. Studies 72, Princeton U. Press, Princeton, 1971. [SGA6] Berthelot, P. : Theorie des intersections et theoreme de Riemann-Roch (SGA6) , Lecture Notes in Math.

G) is a subset of (For more details. see [Hirz. 1J. [Milne. 122]. [Gir. ) For example. the patching data described above for the vector bundle P forms a l-cocyle for ~ with values in 1-1 correspondence between the set rank n vector bundles on X. SL) n of "rank n [St] [Weil]. n If vector bundles on X with structure group P € SP (X). then the patching maps n above. this implies that Proposition: Let An{p) the pointed set SL (U n n V). The group A* and the orbit set SP (X). n to the pointed set of all rank guy for As we observed = OX' X be a scheme.