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Berthelot P., Ogus A. Notes on crystalline cohomology (Princeton, 1978)(T)(264s)_MAh_.djvu |
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isomorphism OT00 DY/SA) * P^ Y(TxY), and proves the proposition...
The top triangle is exactly the cocycle condition fo
The slanted arrows come from the morphisms:
6, \i = (id9<5)oE , v = (s6id)ou ...
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...
This is
because it is not clear a priori that the adjunction map is
compatible with Cech resolution...
The objects of Cris(X/S.)
are pairs Cu,T) such that n 6 К and T e Cris(X/S ); the
mol-'phisms (n',T') -* (n,T) are S-PD morphisms T' -¦ T (and
don't exist unless n' ? n), The covering families of (n,T) a:
just the Zariski coverings {(n,T!) * (n,T)} ...
For each n, there is a natural base-changing
isomorphism:
X
Kr(X/S,E) в An >Kr(X/Sn,i*E ) •...
D
In particular, we obtain the desired relationship be
the crystalline and DeRham Betti numbers...
For the reader's oonv
we recall the following description of the Cartier isomor
proved in [2, 7.2]...
We shall
prove that V is a quasi-isomorphism by unscrewing the gauge e
always relying on the Cartier isomorphism (8.4) and its relation
to ?
8.9 Definition...
;
then we can find a gauge c' > e such that e'(i) = e(i) for
almost all i and such that c1 is steep at j and j+1 whenever
0 <...
That is, there is a c'ommutat
diagram:
FX'fiDY1 V(Z)/S *FX/S*(ni>y Y(Z)/S)n
A ,1 Л,I
FX'nDx, Y(Y)/S * XjY
Moreover, if (Y.F.,) and (Z,F2) are any two lifted situations
which X embeds, then we can also embed X in (YxZ, Fy x F^)
(as a locally closed subscheme, but no matter) — and this
both to (Y,FV) and to B.,F«)...
Now the map j? is just the obvious one:
Ht(Y,pe(s^y/s) ¦* Ht(Y,ny/s), and since ny/s is p-torsion free,
this map is an isomorphism onto pe s H (Yjfly.g)...
The Newton polygon of an F-crystal lies on or above
its Hodge polygon, and both have the same endpoint...
Before doing
i>0 1 n
so, it is convenient to investigate thrse intsrestir.g mapping
properties of the functor r«...
Note that if we take T1 = T, = T in the
above, we can write p p -,(x вТ+х„вТ+г) as S +x (p)R(z), and
hence we get the formula:
S = S S = S S
We view the latter as being, the product of formal power
series with coefficients in the (commutative) A-subalgebra &
of End(P(M,N)) generated by
:' rr ;f...
Now any p e P(M,N) determines 'an A-linear
evaluation map e :?f-* N, sending any D to D(p).@)...
If, moreover Nn has
finitely generated cohomology, it is isomorphic to a complex
M of finitely generated A.-modules, and since each term of
the complex F.6M will involve a uniformly finite direct
sum of terms as in СБ2.2), we see that the arrow is an
isomorphism...
L
begin by recalling the following easy result, of which t
above is "the derived category version"...
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