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Berthelot P., Ogus A. Notes on crystalline cohomology (Princeton, 1978)(T)(264s)_MAh_.djvu |
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Ly(d): Ру/5A)8Пу/5 * pY/SA)enY/S behaves as follows:
Ck,3 [k ] n [k.] [k.-l] [k* ]
Z 5 вш) = a I 51 L ...5i г ...?п n
[к,] [к]
Proof...
(This can be made explicit
from the formula F.11).) Hence we are reduced to proving that
this complex is a resolution of 0™...
?
Using the same technique we can also obtain resolutions of
other crystals of 0^ .g-modules...
Let D = Dx Y(Y) and let F^Eefl^g) be the s
complex of fip/g which in degree q is J "q ЕвП^,„.(where
is the ideal of X in D), Then:
G.2.1)
G.2.2)
Proof...
Moreover, by the quasi-coherence of crystalline
cohomology G.6), we may work over A = PCS," ) instead of
Then G.15) applies, so we need only check that R fy/c E
coherent on S...
For the reader's oonv
we recall the following description of the Cartier isomor
proved in [2, 7.2]...
is a quasi-isc
From the definitions:
so that the arrow induced by dop" is ал isomorphism...
Recall that an
inverse system is said to be "eventually strict" iff the
transition maps are surjeetive for n >> 0...
By induction on r we see that Est(fi'/C) and
E^t(Fef2y/s) are torsion free and that =?r(ny/s) = d^1 (fiy ,/g, ) = 0
for all r ...
= Spec к ; Then
the absolute Frobenius endomorphism Fv of X induces a c-linear
endomorphism T of H* • (X/S), where a is the Frobenius auto-
automorphism of W...
The
eigenvalues of the matrix depend on the choice of basis, but
their p-adic ordinals do not...
B.8.2 The natural maps H^L" ) -*¦ jLim H^L'BA ) are isom
B.8.3 Each H1(L") is a finitely generated A-module...
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