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Berthelot P., Ogus A. Notes on crystalline cohomology (Princeton, 1978)(T)(264s)_MAh_.djvu |
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i i /.о л ,T g
show that the ideal Of X in 0„8Р ._A) has a PD structure com-
compatible with Y ...
There is a natural map: 0y -> PY/SA) ~ |"У@У)
sending x to хв1, which is compatible with HPD stratifica-
stratifications...
Of course, this arrow is the same as our fancy
looking Ь^е changing arrow, and the theorem is proved in the
special case...
Then if E is a
coherent crystal on X, Kf • E is a crystal in the deri
ens a
г я
gory of 0Y,g-modules...
However, in the next chapter, we shall need to
work with the crystalline cohomology of nonproper schemes, and
therefore we have to give a different construction...
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* ...
The Frobenius auto
morphism of W is a PD morphism, covered by the absolute Frobenius
endomorphism F....
is a quasi-isc
From the definitions:
so that the arrow induced by dop" is ал isomorphism...
It seems
reasonable to attempt to define the derived functors of the
s-construction, in the context of the filtered derived category...
Because the arrow * is
independent of the choice of embedding, we can use the lifting
to calculate it, so that C.8) implies that it is a quasi-
isoinorphism...
Assumption (b) implies that Y is sur-
surjective, and the diagram, implies that у is surjective...
It is worth remarking that it depends, in fact, only
on T8idn: М8П ¦*¦ М8П , where ft is the fraction field of W(ic)
In fact a theorem of Dieudonne and Manin [ 3 ] asserts that
the Newton polygon of T8idj...
Using the fact that formation of Г commutes with base
•- n
change, we see that x—i-x is a polynomial function of
weight n...
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...
is an acyclic complex of protective A .-modules and
is bounded above, then there exists an acyclic complex 1С of
projective A -modules, still bounded above, and a surjective
map K* -¦ K* - ...
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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