32 N.M. Katz: is defined (because, in dimension n, all the "lower" ones H-W(/), i<a, have source and target the zero module, hence are isomorphisms). In fact, it is none other than the composite (analogous to (126.96.36.199.3)) (188.8.131.52) 1s,n--a,a ~.~l~n f [r ~ a,n--a cona.~2 Jt~, j, t'~X/S) ~ E1 F~ (R"-"f, (f~x/s)) .........n-w(.) .+ .........................:: R"-"f,(~x/s). (184.108.40.206) Proposition (Igusa, Manin). Assumptions as in (2.3.6), suppose S affine and so small that all the Hodge cohomology sheaves are free (9 s- modules. Let co 1 .....me be a base of R ~- af, (~ax/s) ' and denote by (aij) the matrix in Me(F(S, (_gs)) of H-W(a) with respect to this base: (220.127.116.11) H-W(a) (F~s (coi)) = Z aji o)j. i (18.104.22.168) Consider the dual basis o9",...,o9" of RN+a-nf~(~'~N~sa ). By duality and the definition of the integer a, it is also the least integer with RN+"-"f, (12~7s ~) non-zero, so that the degeneration of the Hodge =~ De
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