# Download The cohomology of Chevalley groups of exceptional Lie type by Samuel N. Kleinerman PDF

By Samuel N. Kleinerman

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Example text

Q. does not divide the order of the Weyl group. -torsion and that in addition, (4:6) holds. Also 11m BG(Fqt) ~ BG(Fp ), the space t whose cohomology we are interested in. Proof of Proposition 4-2. and let q = pd. Q.. , 30 SAMUEL N. KLEINERMAN = {BG(JFqt) I t 1,2,3, ... } is a cofinal system for lim BG(JF t), so can be t used to determine H*(BG(Fp );~/~). p Consider the following diagram. y* H* (B; ~/t) ~ H*(BG(lFp );~/£) H*(BG(JF t) ;tZ/£) q (4: 7) i* ! B* ~ 6* Here n is the rank of G. The maps 6*, i*, and j* are induced by the maps - *xn 6 G( IF \ JF <+ p' , P JF q (]; *xn <+i *xn '-+j G(W q ) , and G(II:) , which can be chosen so that by the naturality of the construction the diagram commutes.

By COHOMOLOGY OF CHEVALLEY GROUPS 35 We must determine the effect of ~q for ~~ coefficients in order to compare the Serre spectral sequences of

I We now use induction. t i On the other hand, since Ker 6*, it must be in the image of some differential. Clearly j for 1-: < This is because eV8ry t occurring in x SAMUEL N. KLEINERMAN 42 will have degree less than the degree of t the d's on such classes by induction. + J l so that the lemma is proved, Using the same argument employed for G we can conclude that 2 (1+1)/2 q ¢ si+l for both LZ and :7Z~-coefficients. sequences for fibrations