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Berthelot P., Ogus A. Notes on crystalline cohomology (Princeton, 1978)(T)(264s)_MAh_.djvu |
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an HPD differential operator, 1„(и): 1„E) •* ly(F) is the na
induced from the stratification on p, ^у/я^' v^a C'*3) *i*
an HPD differential operator using the right structure of i.,,
but is Oy-lir.ear and even horizontal using the left struccur
In summary:
6.9 Construction...
The problem is to
i i /.о л ,T g
show that the ideal Of X in 0„8Р ._A) has a PD structure com-
compatible with Y ...
) which sends Fmn^ <?> to J^ , and a
chain homotopy s: ion - id^ , (the 0™-linear map sending
С d? to $ -* ) under which f^ <?> is invariant...
Then we
get from G.1) a natural isomorphism in the derived category:
SinCe EDV Y(Y)eO«Y/S
" Л, I I A,I
is a bounded complex of quasi-coherent Oy-modules, the
theorem for our X follows from standard considerations...
First observe that there
is a functor L which associates to any A-module E a flat
A-module L (E) and an epimorphism L (E) -» E...
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...
A sheaf
can also be described as a collection of sheav
F € (X/Sn> r^s for all n, together with compatibility maps
as the reader can imagine...
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,
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...
Moreover, if r| is increasing, there
is a canonical quasi-isomorphism: С К' ¦+• (K') , in the notation
of (8.6)...
Then u = y-z lies in pn+ Pn+^Cpe(p)
С peJ^+J , and hence y= ш+z lies in Mn+1 n Pej^^ and li
x, as required...
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...
More-
Moreover, once the arrow * is defined, it "must be a quasi-
isomorphism, because this is a local question, and we may there
fore assume that (X,FV) lifts...
Suppose that K' is a bounded complex of
abelian sheaves on X such that:
(a) H*(X,K#) is p-torsion free...
?
If e is tame, (8.18.4) shows that Fef<y/q is 3uy /s J^.c'.v
and hence, depends only on X-YQ...
To prove the theorem, and perhaps to give some
insight into its meaning, it is helpful to baldly list the
inequalities which it asserts...
The reader
who so desires can also prove the Intermediate statements (mod p for all n),
(8.41) Theorem: With the hypotheses and notation of (8.26):
(8.41.X) For any tame gauge e, the map
Hcri.(Xl/8'*X//3S) - Hcrls(X'/S' V'S> 1S i»J-c"*«...
We want to show that there is a PD structure у on the
ideal r+(M) = ffl Г.(М), with у (xCl]) = xCn]...
Now Гд(М) is the A-subalgebra of ГД,(М') generated
{x :x e M}, so that Lemma C.8) tells us that it is enou
to check that y-(x ) G Г...
This filtrat
* n m n
is compatible with the J-adic filtration, and gr_H (L*) is
finitely generated grTA-module...
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