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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...
Our first goal is the following result, from which all the finite-
ness and base changing properties to come will be deduced:
7.1 Theorem...
We obtain a commutative diagram:
H ^X/bcris' °X/Sh >JH l zar'V Y(Y)/S'
X,T
f*
H*(X''/Scris'0XVS-)- _...
Namely L (E) is
the sheafification of the presheaf which takes any U to
the free A-module with basis ECU) - {0} ...
(The fact that the arrow in the second line is a quasi-
isomorphism depends on the injectivity of the modules j .)
This gives us the desired arrow...
Suppose, in the diagram below, that f is
quasi-separated and smooth, that Y is quasi-compact, and that
X1 = XxyY'...
To define the cohomology we want, we construct a site
Cris(X/'S) which computes the limit automatically...
A convenient way to view an inverse
system indexed by ]N is to make the category К into a site by
endowing it with the coarse topology, i.e...
Then there
л, T
is a D-module with integrable connection (E,V) inducing' E,
and for each m, there is a natural isomorphism:
*их/§* Jx/S
Proof...
It remains only to explain the last statemen
But H^ris(X) - lim H1(X/Sn ,0X/S ) =s lim H1(Y/Sn ,
by G.Ц), which is turn is the same as H1(Y/S ,ft* ...
Now set n = j+?(j) = i(j) = n(j-l), and notice tha
^nY/S^ ' " ^Y/S^n ^or ^ * i-1»3 • * claim that there
commutative diagram:
*J P"n
BY /S C " ZY /S ——- "J(nY /«5
Vb0 Y0/S0 YC/S
which evidently proves that the arrow p...
Since
exact E.27), it preserves images and finite suns, and hence
this is the same as jy^p6 (l)JCl~q]Y.*^/s ) ...
О
The above remarks provide us with a crystalline interpreta-
interpretation of the source of the arrow 4 of (8.8)...
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...
If x € Im(*) Л рг H^ris(X/S), then x = рГу for
some у such that nv(y) 6 ТХ~Т H^(X/S0)...
To
express it, recall that the Newton polygon of an F-crystal
T: M -» M over W is defined as follows: Chpose a basis for *
H, and express T as a matrix, as if it were linear...
A polynomial function f is said to have
"weight n" iff for all r q R and all z e M»AR, fR(rz)= rnf(z),
in which case we write f e P (M,N)
Any f G P(M,N) can be expressed as a locally finite sum
of homogeneous ones...
Actually the hard part of this theorem is the
establishment of the existence of У , which we have already
It is easy to see that if x e M , Ф(х) = ?6 ...
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