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Berthelot P., Ogus A. Примечания относительно прозрачной когомологии (Принстон, 1978) MAh

Berthelot P., Ogus A. Notes on crystalline cohomology (Princeton, 1978)(T)(264s)_MAh_.djvu

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Date Jun 22, 2005

Cites: Thus, we hav
rD л ,Y s >
(jy (Xoy)*t)T =* PT*Py*(H), where py: D^CTx*) - Y is the
natural projection...
To deduce F.14.2), we need to find a nice
quasi-isomorphism:
In fact, there is even an isomorphism of complexes of
L@)-modules: L(E'enA.g) -*• ЕвЬ(ПЛ, ), which, since it is
linear, preserves the ideals К , hence the filtration
To find this isomorphism, we work with the associated HP2-
btratified Ov-racaules, in the following preposition...
How E^ is the cohomology of the oompl
E0 = fX/S<,i(p)crisJ(p)cris(E') and since the comPlex 3(p)eris(
is a resolution of j, ...
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
to triangles)...
We are now ready to study the base changing properties of The
cohomology of a flat crystal...
= p'.: ,
see from G.11) That there is an isomorphism:
Since f: X—* Y is smooth 0R1f .„ E)v is 3R1f;r(?.,0f;' /v
we have exactly an integrable, quasi-nilpotent conrieotio
К fiv(E,.t)fi' /.,) — the Gauss-Manin connection...
Because Hf^^^(E) is bounded above, we can find a complex
L' of flat Og-modules representing it, still bounded above...
Thus, the
complex, as well as its cohomology, satisfies the Mittag-
Leffler condition, hence by [EGA Ojjj 13.2.3], we find that
Hq(lim r(V, Г)) = lim Hq(V,In) = 0 if q > 0,
i.e...
,
ORYftF>n a»r(X/Sn,j*F), and
n
7.22.2 If E is a quasi-coherent sheaf in
then л
Kr(X/S,E) a Klim Kr(X/Sn,i*E),
and
ЖиХ/§*Е ~
*(inE>-
Proof...
In the second part V)e interpret the calculations in terms
of crystalline cohomology and obtain C.20), which does have
global meaning...
is a quasi-isc
From the definitions:
so that the arrow induced by dop" is ал isomorphism...
Consider the topos (X/S)cr^s described in G.17) (recall
that we are writing S for S), and regard the PD filtration
JjJs as a filtration on the object 0x/s (we set j^j = 0X/S if
i <...
From the exact sequence:
о ^ JCi] прп+1"еР -> JCi] -> PeJCi] -> о ,
n n n n
we see that it suffices to check that R1 Ии (Г1-'npn+1"ePn) = С
Since this inverse system is essentially zero, this is clear...
Since Z has finite
projective dimension, this makes sense even if F* is unbound
we can take L"(F*) to be bounded below if F'is...
Consider
diagram:
e(s)
The base changing we just established implies that a is s
jectivej the degeneracy of the Hodge spectral sequence of
X(s)/k(s) implies that e(s) is surjective...
*
Therefore it suffices to prove that И1(Х, Lnn/C +m) has
и m
length _< mh°+" •+hm ...
It seems to be worthwhile to prove a stronger version of
(8.26), which tells us that even over a p-adlcbase, the filtration of crystalline
cohomology provided by ^„/g *8 determined ЬУ the action of Frobenluo...
Let ГЛ(М) = Gt(M)/I,(M), and for each x t M, let x
denote the image of (x,n) in гЛ''5)> If we assign to (x,n)
degree n, the algebra ЕД(М) becomes graded, and the ideal I .(M
a homogeneous ideal...
Expanding in powers of T, we see that ther
are unique elements D (p)D(z) of N8R such that
X К
S (p)-(z) = j/l](p).(z)f ...
Then for each n and N, we have exact sequences:
О -+Рп(М",Ю—> Р^М.Ю^ Pn(M',N)
and
rn(M')z; гп(ю -* rn(M") -+ о ...
To see that $ is a PD morphism, it suffices
check elements of the form x , since these generated Г (М
A
as an ideal...



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