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Berthelot P., Ogus A. Примечания относительно прозрачной когомологии (Принстон, 1978) MAh

Berthelot P., Ogus A. Notes on crystalline cohomology (Princeton, 1978)(T)(264s)_MAh_.djvu

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Date Jun 22, 2005

Cites: LV(E), a crystal on X/S by F.5
ens i
We can give another description of L(E) in terms of our locali
zation diagram E.24)...
Namely L (E) is
the sheafification of the presheaf which takes any U to
the free A-module with basis ECU) - {0} ...
separated, and g: X'-t-X ccvers a PD morphism u: (S ' , 1' ,Y ' ) "¦' CS, I, Y
Let E be a quasi-coherent, flat crystal on X...
Of course, this arrow is the same as our fancy
looking Ь^е changing arrow, and the theorem is proved in the
special case...
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...
The category
(X/S.) ^s of sheaves on Cris(X/S.) can be interpreted as i
E.1): for each (n,T) € Cris(X/S.), a Zariski sheaf- F, T)
on T, plus "compatibility morphisms"
Notice that if n1 < n, we do not require the map
u~ F( T)*^"fn' T) to ^e an isomorph*sm> in general...
Thus, the
complex, as well as its cohomology, satisfies the Mittag-
Leffler condition, hence by [EGA Ojjj 13.2.3], we find that
Hq(lim r(V, Г)) = lim Hq(V,In) = 0 if q > 0,
The Frobenius auto-
automorphism of W is a PD morphism, covered by the absolute Frobenius
endomorphism Fx of X, and it follows that Fx acts on the crystalline
cohomology of X relative to W...
It is clear that what
we must prove j.s that the natural map:
-*lim Ep Ji: J is an isomorphism...
•*• K'9Z/pZ are isomorphisms for г ¦>> 0, and assump-
assumption (a) implies that TH^K') + ЗН'Чк'
is surjective for
all i ...
It is nonetheless apparent that there should be some r
tion between the Hodge filtration and Fv/C, and from (8.23
л/ о
we can also expect the conjugate filtration F" (associat
the spectral sequence Epq = Hp(X,Hq(n^/s ))->ffli(X,U^
play a role...
We regard D as an endomorphism of P(M,N)
and S = У D 1 T1 as a formal power series with coefficients
x i=0 x
the ring of all such endomorphisms...
is apparent from the preceeding paragraph that it suffices t
construct a linear map f:F (M) -»¦ N such that f(x 'L ) = у
whenever |q| = n, (using the same notation)...
Now write P^_^ as a quotien
of a free An-module P^ and take K^ = P^ © P^ with the
obvious boundaries...
Then the given morphism Dn -+ Dn_1 in
D(A ) comes from a morphism of complexes Dn -¦ Dn_^ , unique
up to homotopy...

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