l-adic cohomology

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Again H I (C, III ~ ) =H°(C, 0 ) =k X C C I 2 and H (X, III ) = II (X. 0 ) = k, and since the map H (C, III ) ~ H (X, II ) X X C X is a surjection it must be an isomorphism. So the map HI (X, II ) ~ HI (X, II (Cn .. X 2 ° 0 X is surjection. The regularity of the adjoint system implies that the map 0_ I H (C, II_) --.. H (X. ~) is surjective. ° H (C C II ) , C -+. I So we see that the map H (X. III ) is surjective. The exactness of the sequence X 1 implies that H (X. IIIX (Cl) = o. i. e. I H ex, 0 (-Cn X =o.

So to prove equality one only has to show it locally. So we have a sequence of semilocal rings, Aot:: Al t:: A2 t:: • •• the corresponding morphism of Spec A. 's are finite and birational. J the same letters denote the conductors. ideal of A o • Let a c f. - So a. A. • f - -2 -; 0 t:: f • f t:: Ai Let is an • Since Ai'S are semilocal, relative conductors are principal above, i. e. A where t. belongs to f. tll show that at is an element of the relative conductor of 1 Al c: Ai and by induction we will be through.

Proof. 1. Choose H as before. H nH 1 nC ::~. Let f Thus g is surjective and d 2 :: O. If n ~ X+ 1 then, d n >d n + 1 or d n :: O. Now choose another hyperplane HI such that = 0 and g = 0 be their respective equations. >O 1 1 1o o o The bottom sequence is exact because, f and g hAve no common zeros on C. O ~ ~ p p M(n+1) /' o is exact, ~0 From this we get a complex, HI (J(n-l» ---'> HI (<)p(n-l >EBJ(n)E9J(n» ---+Hl PIP(n) EB qp(n)~J(n+l)). Denote the homology at the middle by Wn + 1 • 2, CLAIM. Wn +1 = Coker (Ho(J(n+l)~~(n)~ HO (Op(n+I)), exact sequences as follows for n ~ We have X + I, o i O--~ Wn +1--+ H1 (M(n+l)) --+H1 (J(n+l)) --+ 0 i H 1 (J(n)'+ J(n» T.

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