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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...
To prove that В is horizontal (i.e., that it preserve
the HPD stratifioations) we have to chase a tedious diagram...
Then there is a
0
Л / О
canonical quasi-isomorphism:
6.14.2 If E is a crystal of Ox/g-modules and if (E,7) is
the corresponding sheaf of Р„ y(Y)-modules with integrable...
•• X is the canonical projection E.18),
there is a natural isomorphism in the derived category of
sheaves of abelian groups on X
G.1.2) *UX/S*<E) - E
Proof...
(Recall that this is the complex which in degree к is
|| Hom[M ,1 ], with the usual boundary maps.) Note that since
i
is bounded above and I* is bounded below, the product "Jf* is reall
i
only a finite one...
Suppose S is quasi-compact,
f: X + (S,I,Y) and f: X ' * (S ' ,1' ,Y ' ) are quasi-compact and...
Of course, this arrow is the same as our fancy
looking Ь^е changing arrow, and the theorem is proved in the
special case...
By induction, IRf^.g E
0 n-1
has coherent cohomology, and the theorem follows from the long
exact sequence of the cohomology of a triangle...
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"
if
Notice that if n1 < n, we do not require the map
u~ F( T)*^"fn' T) to ^e an isomorph*sm> in general...
,
ORYftF>n a»r(X/Sn,j*F), and
n
7.22.2 If E is a quasi-coherent sheaf in
then л
Kr(X/S,E) a Klim Kr(X/Sn,i*E),
and
ЖиХ/§*Е ~
*(inE>-
Proof...
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...
Since
exact E.27), it preserves images and finite suns, and hence
this is the same as jy^p6 (l)JCl~q]Y.*^/s ) ...
Because each jt : Xn -> X
is locally an open immersion, и*Жи„,,, F гШи
n X/Sft
n л'"к x"/sft
for any abelian F ? (X/S) ...
°r-l °r
To prove this, observe that a ^ is a simple augmentat
of a at r , so we have commutative diagrams:
r-1
Н (K*eZ/pZ)[-r]
О >Tr_1(K*9Z/pZ) >Tr(K'eZ/p2) >Нг(К*в2/рЖ)[-г] >О
r-1
Hi(Tr_1)
¦ Hx~r(.HT)
First of all, because K* is bounded, the maps K* ¦+ K' and
Tr(K"8Z/pZ)...
Thus, the map j1: j'E^F^' $) + F*ba(fi" ._) induces an
isomorphism: ,
It follows that j is injective, and (by induction on s), j1
induces an isomorphism
Taking s = i, one has the proposition...
= 0 for j > k, and
in that case Km) implies that whenever m > max{k,n}
maQ + (m-l)a1+-•-+(m-k)ak ^ mbQ +(т-1)Ь1+>••(m-n)bn ...
The reader may wish to note that we have in fact proved a
slightly stronger inequality than claimed, which he can work
out according to his needs...
Then for each n and N, we have exact sequences:
О -+Рп(М",Ю—> Р^М.Ю^ Pn(M',N)
and
rn(M')z; гп(ю -* rn(M") -+ о ...
B.8.2 The natural maps H^L" ) -*¦ jLim H^L'BA ) are isom
B.8.3 Each H1(L") is a finitely generated A-module...
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