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Berthelot P., Ogus A. Notes on crystalline cohomology (Princeton, 1978)(T)(264s)_MAh_.djvu |
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Size 2.0Mb Date Jun 22, 2005 |
the HPD "Stratification on 0 corresponds to the identity jnap
Y/S^^ "*" ^Y/S^^' an<^ *^а* tne one on Py/c^' corresponds to
the map Ру/8A)вРу/5Ш - Ру/5A)8Ру/3( 1) coming from
a8b8c8d > ledeaebc .) It follows immediately from the
local coordinate calculation F.11) that the composition
!)y "* LyCOy) -+ LyCn^.g) is zero, and hence that we have a morphis
of complexes 0
We get induced a map of complexes of crystals on
Y/S: 0y/o * ^уСу/Б^ an<^ hence by applying i ...
Then bf*E' is f*L
К Hom^tOLf*E#,F") is Нотд,(f*L#,1*), and Ж Нотд(Е',lRfftF') i
Horn'(E*,J°)...
But this is just Нот.(L8P*,J), and since L is flat and J is
injective, it's just a resolution of Нот.(L8B,J), and in
A
particular is acyclic...
Then as in the proof of the previous re
if K* ~is"~acyclic in degrees < m, one has a quasi-isomorph
P- -> A', with' АШ = p'VdCp), An = 0 if n < m, and An
if n > m...
We
want to construct crystalline cohomology of X over S, compatible
with the cohomology of each ^^sn^cr±s • For each m ? n, we
have a morphism: imn: (X/Vcris + (X/Sn)cris ' comPatible
with composition...
Let us remark
(BID shows that in the proper case only, KI4X/S,E) is f
torially determined by the inverse system KHX/S , i* E )
7.25 Corollary...
Then Lemma (8.11) implies that 4? is a
quasi-isomorphism, and the proof of Theorem (8.8) is therefore
complete...
It is easy to see that if c! is Deligne's "filtration
canonique" [1, 1.4.6], there is a canonical filtered quasi-
isomorphism с!к* ->C.K" ...
there is a commutative diagram as shown, in which У is an
E
x/s
'FX/S*KuX/S*°X/S
This diagram is functorial in X and e , and agrees with the
diagram (8.8) in a lifted situation...
Since
I С р П' , this implies that on cohomology, Fv ,
maps H1(YI ,Fefjyt/s) into pe M^CY.fJ^g) - provided, о
course, that e is tame...
If, additionally, the Hodge spectral sequence of X1
q
[e 3
[
generates at E.^ the map ar: K*(X'/S ,JX,
is injective...
Bv
a well known argument, it suffices to prove that the functor
on finitely generated A-modules:
?: M «-> lim (Т
is exact...
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