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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: Lv construction furnishes us with a large collecti
of HPD stratifications, and hence with a large collection of
crystals, via F.6)...
To see that we get
a complex Ly(fiy^s), the reader can show that the composition
* "y/S "* *V/S *s zero as an ^PD differential operator...
Recall that if F i
any Oy-module, L(.?) = Р„,дA)вл F , and
e,: PA)8L(F) ¦* L(F)9PA) is deduced from the map:
(авЬ) 8 (c8d) 9 x -* (led) в х в а в be
flatness of the P's and the adjointness of 8 and Нот show
that ffomA(P",J) is a complex of injective A-modules, in fact
an injective resolution of WomA(fjkB,J)...
The category
(X/S.) ^s of sheaves on Cris(X/S.) can be interpreted as i
E.1): for each (n,T) € Cris(X/S.), a Zariski sheaf- F, T)
on T, plus "compatibility morphisms"
Notice that if n1 < n, we do not require the map
u~ F( T)*^"fn' T) to ^e an isomorph*sm> in general...
Moreover, the inverse systems H^X/S ,i*E) satisf
ML, and there is an isomorphism:
H1(X/S,E) -» lim H1(X/Sn,i* E) ...
Both for the proof and applications, we shall in fac
to make a more general statement, ( 8.8 >...
Then Lemma (8.11) implies that 4? is a
quasi-isomorphism, and the proof of Theorem (8.8) is therefore
Consequently, we
deduce an isomorphism:
Now recall that the qth term of
Ip?(i)FJ.LD/b_) : ?p?(i)j^<J[i-*Vn§/s) ...
The above remarks provide us with a crystalline interpreta-
interpretation of the source of the arrow 4 of (8.8)...
Moreover, once the arrow * is defined, it "must be a quasi-
isomorphism, because this is a local question, and we may there
fore assume that (X,FV) lifts...
Suppose that K' is a bounded complex of
abelian sheaves on X such that:
(a) H*(X,K#) is p-torsion free...
There is a
natural morphism of filtered-complexes : j: (Fefly .g ,F*)-<- (Юу -s ,F" )
and hence a morphism of spectral sequences:
3r • br u "y/S'F ' г UiY/S' ; *
Let S1 = S*<, _ SpQ , and observe that Jeid0 is an iso-
find a relationship
the form of inequalities, between the p-adic divisibility prop-
properties of Ф and the Hodge numbers of X...
Then if M
A-module, we have by (A3) that ГД(М) а Гр(М)врА, and hen
there is an exact sequence: Г (M)9 I ¦* Гр(К) * Г^<М) -* 0
must show that 1Гр(М) П Гр(М) is a sub PD-ideal of Гр(И
Clearly 1Гр(М) П Гр(М) = 1Гр(М) is generated by the set
elements of the form ax, where a e I and x € Гр(М)...
Appendix В
Finiteness of
In this appendix we will prove the finiteness theore
lUim which we need in §7 in the proof of finiteness of P-
crystalline cohomology...
Since we have Blim D»AQ & DQ,
H1(KLim D)8.A0—S-*H1(D(.), hence is finitely generated, hence
so is H^KjLim D)...
For each i, let B1+, be the image of d1":
Ег~Г * Er ' and let 5r+l be the Preirca8e of 3i+i in E2
Then В С В +...

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