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Berthelot P., Ogus A. Notes on crystalline cohomology (Princeton, 1978)(T)(264s)_MAh_.djvu
Date Jun 22, 2005
rD л ,Y s >
(jy (Xoy)*t)T =* PT*Py*(H), where py: D^CTx*) - Y is the
It is easy to see that this set is a sub PD ideal
of KTm • Hence it suffices to check that it contains a set of
PD generators of Kl, , for instance F...
(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) ...
Since the modules comprising L'(E' ) are
flat, Lf* = f*L" takes acyclic complexes to acyclic complexes,
hence quasi-isomorphism to quasi-isomorphisms, (and triangles
The key fact of cohomologica
descent, which we shall use without proof, is that for any
abelian sheaf E, the natural map, E —>-IRirftTt*E is an isomorphism
Using this, we can reduce our proof to the topos ^"/^^cris
Construct a topos (Х''/S1) ...
There are several
ways to do this; we hope that the reader does not find the one
we have chosen too distasteful...
Now the base changing theorem (
implies that the arrows D 9 A , ¦+ D , are (quasi-)
isomorphisms for all n, so that D is what Appendix В с
a "quasi-consistent complex of A.-modules" (c.f...
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, (8.20) -cells --is that
relative Frobenius Fy .,, induces a map (in the derived category
n(i) = e(i)+i ...
Now this spectral sequen
is, after renumbering, the spectral sequence of the canonical
filtration [1,1.4] T...
(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...
Now any p e P(M,N) determines 'an A-linear
evaluation map e :?f-* N, sending any D to D(p).@)...
Finally, it is clea
J- у I1 ¦ •¦ Л* Лш J1 Л,
that iji is unique, because У(М) = Г (M) generates Г.(М) а
has finite tor-dimension, so there exists an m such
that Hx(Dne (МвдАп)) э< H1(FneA M) = 0 for all i < m and
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