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Berthelot P., Ogus A. Notes on crystalline cohomology (Princeton, 1978)(T)(264s)_MAh_.djvu
Date Jun 22, 2005
maps from DXjY(TxY) to T and to Dx уШ» respectively...
We obtain a commutative diagram:
H ^X/bcris' °X/Sh >JH l zar'V Y(Y)/S'
(Recall that this is the complex which in degree к is
|| Hom[M ,1 ], with the usual boundary maps.) Note that since
is bounded above and I* is bounded below, the product "Jf* is reall
only a finite one...
Now let Sn С S be the subscheme defined by Kn+1, so th
we have an exact sequence:
(ft) 0 - Kn/Kn + 1 - 0 _ 0 -» О
Tensoring with the complex Kfy.^ (E) in the derived catego
and using the base change isomorphisms
for v - n and n-1, we get a triangle:
Pfx/s* E *fx/s*E
Now the top of the triangle is isomorphic to ^fy/s E80 10^'Г0 '
again by base changing, which has coherent cohomology by the
case m=0 since Hf-^^ ftE is perfect...
(In the presence of torsion, this may only define a preshea
and K^ is the associated sheaf.)
It is clear that a morphism f: A" •* B" induces a morphism
fe: Ae*Be for a11 E ' and tnat this defines a functor...
Since the value of e(i) for negative i
has no effect, we may assume that e(i) = e@) - i for all i >...
It follows that the arrow
V *V/S«JX/S * FX/S* ^"x/sA/S in the derived category
that we have defined is independent of the choice of embedding...
If e is tame, (8.18.4) shows that Fef<y/q is 3uy /s J^.c'.v
and hence, depends only on X-YQ...
who so desires can also prove the Intermediate statements (mod p for all n),
(8.41) Theorem: With the hypotheses and notation of (8.26):
(8.41.X) For any tame gauge e, the map
Hcri.(Xl/8'*X//3S) - Hcrls(X'/S' V'S> 1S i»J-c"*«...
More precisely, for each A-algebra R, let
*n R:M0R—vT (MHR be the map taking x S M9R to the image
x^n3 in r(M)9R via the inverse of the base change isomorphi
Г (MHR + Г (M8R)...
Then the given morphism Dn -+ Dn_1 in
D(A ) comes from a morphism of complexes Dn -¦ Dn_^ , unique
up to homotopy...
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