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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...
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...
Suppose S is quasi-compact, f: X + (S,I,Y) is
quasi-compact and quasi-separated, and fV/o: (X/S) ...
If E' and F" are two such complexes, the
natural map L'(E') IB L* (F-) -* L'(E*®F') is hence a quasi-
isomorphism, and since L" is also compatible with shifts, one
see that if С(u) is the mapping cone of a morphism of com-
complexes u: E' -¦ F', then there is a natural quasi-isomorphism
C(L'Cu)) ->L'CCCu))...
Of course, this arrow is the same as our fancy
looking Ь^е changing arrow, and the theorem is proved in the
special case...
Then as in the proof of the previous re
if K* ~is"~acyclic in degrees < m, one has a quasi-isomorph
P- -> A', with' АШ = p'VdCp), An = 0 if n < m, and An
if n > m...
We sha
say that a sheaf of ^v/q -modules is a "crystal" iff the map
( T) "* F(n'' T1) are isomorphisms f°r every u...
For each n, there is a natural base-changing
isomorphism:
X
Kr(X/S,E) в An >Kr(X/Sn,i*E ) •...
Since
e-(k'-l) > c(k'), this can only be because e(k'+l) < c(k
Now we can repeat the argument, and we obtain the double
absurdity: e is strictly decreasing beyond к and
e * e1 beyond к ...
;
then we can find a gauge c' > e such that e'(i) = e(i) for
almost all i and such that c1 is steep at j and j+1 whenever
0 <...
It is easy to see that if c! is Deligne's "filtration
canonique" [1, 1.4.6], there is a canonical filtered quasi-
isomorphism с!к* ->C.K" ...
There is just one more unpleasant point to check, that
^Ха^Х11!) /S ^ ~ ^XnD/S " This is not quite obvious, because
TTX* does not preserve images and sums...
Since TD/S is a
morphism of complexes, (8.21.1) follows, and (8.21.2) is ал
immediate consequences...
Thus, the map j1: j'E^F^' $) + F*ba(fi" ._) induces an
isomorphism: ,
It follows that j is injective, and (by induction on s), j1
induces an isomorphism
Taking s = i, one has the proposition...
To see this, form fRrTi:M8 R[T] -* Ne.R[T],
and if x e KBR, write fR[T](x0T) = ][ f^(x)Tn, with
f^(x) G N8AR • (In the sum, only finitely many terms are non-
R A
zero, for each x ...
If F' -*
is a quasi-isomorphism of complexes of flasque sheaves,
rCIH,F') ¦* rON,G') is a quasi-isomorphism...
q
Our aim is to compare the categories D(U,A.) and D(A)
We have defined a functor F.lim: D(TJ,A.) -*D(A), and there
an obvious functor back...
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