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Berthelot P., Ogus A. Notes on crystalline cohomology (Princeton, 1978)(T)(264s)_MAh_.djvu |
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immersion and Y/S is smooth, we shall use the following nota
If 5 is a sheaf of (L-modules, L,(E) will be the sheaf of
0Y-modules with HPD stratification indicated above...
Recall that
Since d is a differential operator of order one, we have a
diagram:
к 68idn к k+1
In this diagram» Lv(d) is the composition of the two horizontal
arrows, w: PV.<,A) ¦*¦ Py/o is the natural projection, and
: PY/S8nY/S + nY/S is the °Y"linearization °f d# N°W since
6 is a PD morphism, we compute:
/ n \
Recalling that w kills ? if I Jt I _> 2, we see that
n r j^_ i»l fir T
idnew9ido maps this to a • I ? x в?.8ш + a?L вш ...
О
' Using the filtered Poincare' lemma and F.10.1), we can pr
a more precise result:
7.2 Theorem...
We begin with a very special case, assuming that X
lifts to a smooth affine scheme Y over S...
We sha
say that a sheaf of ^v/q -modules is a "crystal" iff the map
( T) "* F(n'' T1) are isomorphisms f°r every u...
For each n, there is a natural base-changing
isomorphism:
X
Kr(X/S,E) в An >Kr(X/Sn,i*E ) •...
The above results generalize somewhat the work pf Mazur [U,5
Our technique of proof is, however, rather different,since ws foil
a suggestion of Deligne, proving a local result of which the above
global statements are formal consequences...
One more cchomology computation will complete the proof of
gain, the Cartier isomorphirns plays the key role...
Assuming that H (f) is an isomorphism:
i) H1(f ) is injective: Any class in H1(AJ) is repre-
represented by some p a, with da =0, since A1
is torsion free...
It follows that the arrow
V *V/S«JX/S * FX/S* ^"x/sA/S in the derived category
that we have defined is independent of the choice of embedding...
find a relationship
the form of inequalities, between the p-adic divisibility prop-
properties of Ф and the Hodge numbers of X...
(In particular, the endpoir.t of the first lies to the left of
the endpoint of the second.)
Proof...
D
There is one more fact that I would like to explain,
namelv, the relationship between Г and S :
П -4
(A10) Proposition...
q
Our aim is to compare the categories D(U,A.) and D(A)
We have defined a functor F.lim: D(TJ,A.) -*D(A), and there
an obvious functor back...
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