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Unformatted text preview: and by (2.3.1.2) the formation of its E 2 term commutes with arbitrary change of S'~ S. The result follows by (2.2.1.11). (2.3.3) We are now in a position to explain the terminology "conjugate" spectral sequence. With the assumptions of proposition (2.3.2), suppose further that S is the spectrum of a field K of characteristic p, and that the divisor D is void. The Hodge ~ De Rham spectral sequence (2.3.3.1) E~' b = H b (X, ~X/K) =*" Ha + b (X, ~"~'/K) being degenerate at Et, we have (2.3.3.2) g~ (H n (X, f2~/r ) ) ~ H" ~ (X, f~a'/~) The degeneracy at E2 of the conjugate spectral sequence (2.3.3.3) co,E~2'b =F~H~(X, f2~,/~) =~ H~+~(X, f2~/r) which we prefer to rewrite as (2.3.3.4) r =H~(X ~p~, O~c~/K) =~ Ha+b(X, f2~//~), gives an isomorphism (2.3.3.5) g~on n~ (X, f2~c/K ) ~ H a (X ~p~, f~}?~,~/K) Putting together (2.3.3.5) and (2.3.3.3) (for X tp~) we find an isomorphism (2.3.3.6) g~o H"(X, f2~/r)"~g~~ H~(X ~v~, I2],~,/r ) ....
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 Fall '11
 NormanKatz

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