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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 |
isomorphism OT00 DY/SA) * P^ Y(TxY), and proves the proposition...
(The reader may easily check this from the fact that
the HPD "Stratification on 0 corresponds to the identity jnap
Y/S^^ "*" ^Y/S^^' an<^ *^а* tne one on Py/c^' corresponds to
the map Ру/8A)вРу/5Ш - Ру/5A)8Ру/3( 1) coming from
a8b8c8d > ledeaebc .) It follows immediately from the
local coordinate calculation F.11) that the composition
!)y "* LyCOy) -+ LyCn^.g) is zero, and hence that we have a morphis
of complexes 0
We get induced a map of complexes of crystals on
Y/S: 0y/o * ^уСу/Б^ an<^ hence by applying i ...
) which sends Fmn^ <?> to J^ , and a
chain homotopy s: ion - id^ , (the 0™-linear map sending
С d? to $ -* ) under which f^ <?> is invariant...
Now if E" is a complex of A-modules, L*(E") is a double
complex of flat A-modules, and the associated simple complex
L'(E") is easily seen to be naturally quasi-isomorphic to E if
E" is bounded above...
The idea of, cohomological descent is to replace the topos
(X/S)crig by a topos (XYS) which stands for a finite covering
(U.) of X together with all the gluing data...
Notice also that the above diagram makes
Y0/S0 Y0/o0 Y0
sense for any Y/S, not necessarily smooth...
Ц We shall be especially interested in the case where A
is the abelian category of complexes of an abelian category...
We have a morphisin of filtered com-
complexes in (X/S.) y': Jy/c ""*¦ ^x^^Y/S ^' anc* ^ence ^ог апУ е »
a morphism of complexes: у : J>j/S "* FxL^fiY/S '' To Prove
that it's a quasi-isomorphisra, we may restrict to (X/S )
Now look again at the proof of the filtered Poincare'lemma'
F.13)...
Then u = y-z lies in pn+ Pn+^Cpe(p)
С peJ^+J , and hence y= ш+z lies in Mn+1 n Pej^^ and li
x, as required...
We can easily
find an.open covering X -• X such that Xе-*'Y with
(У ,F „) a lifted situation: Then
Xn = X°xx...xOc- Y°xsY°...Y° - Yn is locally closed, and
(Y , Fy*s,..Fy) is a lifted situation...
, .k(s) (via the absolute Frobenius of
k(s)) we see that
HP(X'(s), n^,(s)/k(s)) *k<6)ek(s)HP<X<8), n^(s)/k(s)) • Thus,
EPqcQn(X(s)/k(s)) has the same dimension as EiqKod e(X(s)/k(s)
Counting dimensions shows that the conjugate spectral sequenc
of each X(s)/k(s) degenerates at Е„...
If x € Im(*) Л рг H^ris(X/S), then x = рГу for
some у such that nv(y) 6 ТХ~Т H^(X/S0)...
To prove the theorem, and perhaps to give some
insight into its meaning, it is helpful to baldly list the
inequalities which it asserts...
To
express it, recall that the Newton polygon of an F-crystal
T: M -» M over W is defined as follows: Chpose a basis for *
H, and express T as a matrix, as if it were linear...
(There are several excellent reasons for calling
them "modular functions," but I shall resist the temptation to
do so.) Here is a fancy description: If M is an A-module,
let M denote the corresponding quasi-coherent sheaf on the
"big" Zariski topology (Spec AJAR of all A-schemes...
In essence, the previous lemma has done this for us: We
have an A-linear map S:M -*• ехр(Л , and hence by (Al), an
A-algebra homcmorphism S:F.(M) -»¦ J? , sending x q to D...
More generally,
is a site and A is a sheaf of rings on X, D(X,A) will be
derived category of sheaves of A-modules.....
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