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Berthelot P., Ogus A. Notes on crystalline cohomology (Princeton, 1978)(T)(264s)_MAh_.djvu |
Size 2.0Mb Date Jun 22, 2005 |
recall from E.25.2) that if T E Cris(X/S),
(jo QoyO*E) = pT ((XoV)*E)p , where p • DY Y(TxY) + f and
p • Dv (TxY) -> Y = Dv Y(Y) are the natural maps...
By
exercise F.5), L^y/S^T ~ h*( L@Y/S)Y) ~ h*(LY(°Y/S) J' Assume
further that h(T) is contained in an open set of Y on which
x1 xn are local coordinates...
We shall leave F.14.1) to the reader, who needs
only transcribe the proof of F.13), replacing 0„ everywher
by M...
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...
This holds for any
morphism of ringed topoi, as explained in [SGAUXVII], where it
is called "trivial duality"...
(The fact that the arrow in the second line is a quasi-
isomorphism depends on the injectivity of the modules j .)
This gives us the desired arrow...
Of course, this arrow is the same as our fancy
looking Ь^е changing arrow, and the theorem is proved in the
special case...
Now the base changing
Ь
theorem tells us that Rfy/s ^E^ в ''s' 2Sfx'/Sl'?'^ a c0Plex
which we can bound (independent of S*) by G.6)...
Its objects
are PD thickenings (U,T,6) as in §5, such that there exists an
n with PnC)_ = 0; its morphisms are defined in the usual way...
is a quasi-isc
From the definitions:
so that the arrow induced by dop" is ал isomorphism...
•*• K'9Z/pZ are isomorphisms for г ¦>> 0, and assump-
assumption (a) implies that TH^K') + ЗН'Чк'
is surjective for
all i ...
Moreover, (8.20) -cells --is that
relative Frobenius Fy .,, induces a map (in the derived category
n(i) = e(i)+i ...
The reader
who so desires can also prove the Intermediate statements (mod p for all n),
(8.41) Theorem: With the hypotheses and notation of (8.26):
(8.41.X) For any tame gauge e, the map
Hcri.(Xl/8'*X//3S) - Hcrls(X'/S' V'S> 1S i»J-c"*«...
XV], but we
shall not need this result.)
We prove (B2.1) by constructing the complex F^ induc-
inductively...
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