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Berthelot P., Ogus A. Notes on crystalline cohomology (Princeton, 1978)(T)(264s)_MAh_.djvu |
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'¦ * *
arrow jy*LE) -*• (Xo/) E a y*(X*(E))...
The top triangle is exactly the cocycle condition fo
The slanted arrows come from the morphisms:
6, \i = (id9<5)oE , v = (s6id)ou ...
Applying the above special case to
(E,V) on Y/S, we see.that there is a canonical isomorphism:
Sinoe icrie4E a K
Ж12агАЖиХ/8АЕ' and since ЧагА is
theorem follows...
This holds for any
morphism of ringed topoi, as explained in [SGAUXVII], where it
is called "trivial duality"...
We begin by factoring f into two morphisms:
(T'.A1)—S—<-(T',f(A))—JL-»(r,A) (the reader can, I hope, eas
imagine the meaning of the middle term)...
To deduce the general case it is necessary, unfortunately,
to resort to the technique of cohomological descent...
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 ...
By induction, IRf^.g E
0 n-1
has coherent cohomology, and the theorem follows from the long
exact sequence of the cohomology of a triangle...
Since uv/§o J = UX/S ' we ^ave a canonical iso-
morphism: Kux/gft 3Rjftj *(J^ E)"^Kux/s ft (j * ^m] ) (
tells us that IRjftj*(JCm^ E) a J^ml E, and one sees easily
* X/S X/S
that j*(J[m^ E) st JCm:i (j*E), and that j*E is a crystal on
X/S X/S...
It remains only to explain the last statemen
But H^ris(X) - lim H1(X/Sn ,0X/S ) =s lim H1(Y/Sn ,
by G.Ц), which is turn is the same as H1(Y/S ,ft* ...
j ? dim(Y/S):
dim Y/S
(This would be quite impossible if we were dealing exclusively
with tame gauges, and this is the essential reason our proof is
simpler than the original one.) It follows from (8.13) thait
Ч1 .Qidj/ z is a quasi-isomorphism, hence from (8.14) that *E ,
is a quasi-isomorphism...
Specifically, if f and g are mcrphisir.s о
filtered ccnplexes (A*,F)—>CS',G), and if R is a filtered
homotopy f - g, then R induces, in the obvious way, a hcmotopy
Re between the morphisms fc,ge: FA'* FeB* ...
filtration
8.18.4 Suppose e is tame, and Y/S is smooth and X = YQ,
тЬлп "Rn 7=s*F0* ~ F* О * — (tl * ^ TVi Чсз +й11о не
¦* * t ^ л V / Qj V / ^ ~™ V V / 4 V V /Q V/Qc * illlb LCllo Uo
that the p-adic interpolation of the Hodge filtration У defined
by the tame gauge e depends only on X, not У...
That is, there is a c'ommutat
diagram:
FX'fiDY1 V(Z)/S *FX/S*(ni>y Y(Z)/S)n
A ,1 Л,I
FX'nDx, Y(Y)/S * XjY
Moreover, if (Y.F.,) and (Z,F2) are any two lifted situations
which X embeds, then we can also embed X in (YxZ, Fy x F^)
(as a locally closed subscheme, but no matter) — and this
both to (Y,FV) and to B.,F«)...
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...
If XVk is smooth and proper, and
M = H*ris(X/W)/(torsion), then the Newton polygon of
lies on or abgve the Hodge polygon of X/k ...
Then a
polynomial function M -*N is just a morphism of the underly- -
ing sheaves of sets M -*• N^ ...
By Theorem (A5), we see that there
is a unique A-linear map j? : Г (М) + A such that r°i = jf ,
i.e., such that j?(xi'n-b = ^(x)....
(B.I.4) A sheaf F on U is flasque iff the inverse syst
F is strict, i.e., iff all the maps F -+ F , are surjectiv
If F -» G is an epimorphism and F is flasque, so is G ...
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