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Berthelot P., Ogus A. Notes on crystalline cohomology (Princeton, 1978)(T)(264s)_MAh_.djvu |
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tion for Eg «
First we must give a more explicit formula for the HPD-
stratification of the L-construction...
Suppose S is quasi-compact, f: X + (S,I,Y) is
quasi-compact and quasi-separated, and fV/o: (X/S) ...
Г) ..Ли-
X0 Lp
Since X/S is quasi-separated, sets of the above form are
not necessarily affine, but are quasi-compact...
(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...
Some additional hypotheses will-allow us to assert that
the base changing map above is actually an isomorphism...
The above argument works only for quasi-coherent M, but if M
Ь
is arbitrary, it implies that the stalks of ]Rfy ,g (E)8/.l are uni-
Ъ
formly bounded, hence so is 3Rfv/c (E)8M ...
To define the cohomology we want, we construct a site
Cris(X/'S) which computes the limit automatically...
If F€ Cris(X/S.), we get an inverse system F.,^ T
of sheaves on T by
Fn,(U,T,6) = F(n,(U,T,6)) = (jnF)(U,T,<5)
(for n >> 0)
To verify that lim F^ .ц т ...
Then:
7.24.1 There is an object D in the derived category of
inverse systems of A-modules such that each D =* Er(X/S ,i*E)
and Klim D sEr(X/§,E)...
For the reader's oonv
we recall the following description of the Cartier isomor
proved in [2, 7.2]...
It will turn out only to have crystalline meaning
for what are called "tame gauges", which we shall discuss later...
It seems
reasonable to attempt to define the derived functors of the
s-construction, in the context of the filtered derived category...
i) H <fn) is surjective: Any class in H1(B") is repre-
repreince
sented by some p° b , with b € в*, db =0...
Because each jt : Xn -> X
is locally an open immersion, и*Жи„,,, F гШи
n X/Sft
n л'"к x"/sft
for any abelian F ? (X/S) ...
If M and N are A-modules, a "polynomial
function f:M -¦ N" is a compatible collection of set naps
{fR:N6AK—+ NQAR} for all A-algebras R...
Recall that we arc- regarding 5 a", an ¦.•Ic.m'p;;i:
•JiCTj], given by Sy[T] = ED^T1, i.e., if p «? РСМ.Ю,
S CT](p) = S (p) ? P(M,N8A[T])...
(B.1.3) The derived functor of lim = ГШ, ) is defined
all of DOHjA.), not just on D QN,A.)...
This filtrat
* n m n
is compatible with the J-adic filtration, and gr_H (L*) is
finitely generated grTA-module...
Let D be its image in D(li,A.);
By (B6) D e D~CN,A.) and is consistent, hence
by (B.7), Klim D € Dft(A) • Jt i-s clear from (B.5...
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