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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: 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 ...
If u: P(l)en ¦* П' is an HPD differential operator
and if, Pg(u) is the induced operator
РA)8Е8П + Е8П' D.8), then 6 takes L(p?(u)) i
Proof^ Recall that L(E) = Py/s^De^ E , so that В i
supposed to be an isomorphism: P(l)8E6fJ -¦ ЕвРA)вП ...
(Recall that the complex
ESpftl .„ is the complex ?8^ fiy/S ' °btained from the connection
on E as 0^-module.) Then there is a canonical isomorphism:
If uv/o: (X/S) ...
(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...
[EGA IV 18.1.2]
Since X is quasi-compact, finitely many of these neighborhoods
will do, and since X is quasi-separated, any intersection U of
them is quasi-compact...
For each n, the crystal E .on Cris(X/S ) с
ponds to a P = V v(Yx S )-module with integrable conne
П Л , i оП
(E ,7), with En = ED , by F.6)...
Cohomologie cristalline des sch
de caracteristique p > 0" Lecture Notes in
Mathematics No...
Then Lemma (8.11) implies that 4? is a
quasi-isomorphism, and the proof of Theorem (8.8) is therefore
complete...
of this complex is
It is clear that these sheaves satisfy the hypotheses of G.20)
hence are acyclic for it..: МхХ +Х ...
Then a duplication of the calculation (8.10.1) shows that there
is a natural triangle:
0 -»1ц' -* Ln -+ «•'(о в idi/ 2H-J] -> 0 ...
Since taking cohomology commutes with flat base change
it follows that js 8id_ is an isomorphism for all s,t,r ...
This is the continuous graph consist-
consisting of the straight line segment of slope 0 over the interval
[О.а.З, of slope 1 over the interval [a...
If M and N are A-modules, a "polynomial
function f:M -¦ N" is a compatible collection of set naps
{fR:N6AK—+ NQAR} for all A-algebras R...
Therefore There
linear f: Гд (Ю+М1 sending x P to x' where x and
bases of M and M', respectively...
To see that $ is a PD morphism, it suffices
check elements of the form x , since these generated Г (М
A
as an ideal...
In fact, one sees easily by descending in-
induction that the complex K^_1 splits: K^-l = Pn-1 6 Pn-1 '
with the obvious boundary maps...
Thi
that L* satisfies the hypotheses of the following Lemma,
therefore will prove the Proposition...



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