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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: This gives us the desired extension, establishes the iso-
isomorphism OT00 DY/SA) * P^ Y(TxY), and proves the proposition...
The essential point of the calculation is the cocycle condi
tion for Eg «
First we must give a more explicit formula for the HPD-
stratification of the L-construction...
On the other hand, E1 = g*E is a crystal on X1,
and :EfAt,eiE'is a bounded complex on S1, which is reasonable to
compare with Lu*B
In fact it is easy to obtain а шар (in the derived category
Lu*IR fx/s E +Kf',.g,E1, i.e...
Of course, this arrow is the same as our fancy
looking Ь^е changing arrow, and the theorem is proved in the
special case...
But Kfy ,g (К)ЭМ is
a direct summand of Rf^,-s (Е)вОд,, and hence it too is uniformly
bounded...
There is a natural morphism of complexes:
JX/S E "* FXL(E-8fiD /S5 where the latter in
V*E.8flg /s) (c.f...
Then Lemma (8.11) implies that 4? is a
quasi-isomorphism, and the proof of Theorem (8.8) is therefore
complete...
It is easy to see that if c! is Deligne's "filtration
canonique" [1, 1.4.6], there is a canonical filtered quasi-
isomorphism с!к* ->C.K" ...
Since Z has finite
projective dimension, this makes sense even if F* is unbound
we can take L"(F*) to be bounded below if F'is...
•'•The reader who so desires can now skip to Katz's conjecture, p
Here we shall develop only those properties of this structure
we need for the applications, and refer the reader to Mazur's
papers for more details...
There is a
natural morphism of filtered-complexes : j: (Fefly .g ,F*)-<- (Юу -s ,F" )
and hence a morphism of spectral sequences:
3r • br u "y/S'F ' г UiY/S' ; *
Let S1 = S*<, _ SpQ , and observe that Jeid0 is an iso-
isomorphism...
in the lemmas which follow, v.-e assume only that >;/sQ i&
and smooth, and state the additional hypotheses as we nee
If X does satisfy all the hypotheses of the theore
base changing theorem for crystalline cohomology and the
ness assumption show that H*rig(X/SHo 0s =sH*R(X/SQ), an
3 0
particular, the latter is locally free...
(X'/S) ¦»• H^ is^/s> deters
the (mod p) Hodge filtration of X'/SQ and conjugate fil
of X/?p, assuming the stated degeneracy and torsion hypo
Even without These hypotheses we car...
Plot the points
(i, ord (a.)); then the Newton polygon of T is th-з convex hull
of this graph...
We want to show that there is a PD structure у on the
ideal r+(M) = ffl Г.(М), with у (xCl]) = xCn]...
Now any p e P(M,N) determines 'an A-linear
evaluation map e :?f-* N, sending any D to D(p).@)...
Then if M
A-module, we have by (A3) that ГД(М) а Гр(М)врА, and hen
there is an exact sequence: Г (M)9 I ¦* Гр(К) * Г^<М) -* 0
must show that 1Гр(М) П Гр(М) is a sub PD-ideal of Гр(И
Clearly 1Гр(М) П Гр(М) = 1Гр(М) is generated by the set
elements of the form ax, where a e I and x € Гр(М)...
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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