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Berthelot P., Ogus A. Notes on crystalline cohomology (Princeton, 1978)(T)(264s)_MAh_.djvu |
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map Oj,ys -*¦ L@y) sends ^x/S ^nto ^ > we see that we have an
augmentation:...
(Leaving out 0
insures us that L takes zero maps to zero maps.) It follows
that there is a functorial complex L"(E) of flat A-modules
with Н^ЧЬ'СЕ)) = 0 if it 0, and H°(L'(E)) naturally ison.orphic
to E...
(The fact that the arrow in the second line is a quasi-
isomorphism depends on the injectivity of the modules j .)
This gives us the desired arrow...
Suppose that in the above diagram, S is quasi-compa
fQ is smooth,quasi-compact, and quasi-separated, u is a PD morphi
(S'.I'.Y1) -¦ (S,I,Y), and SQ С S and Sj С S1 are defined by sub
PD ideals of I'and I1, respectively...
tells us that the map
rv**fw/T,(El(w/T>eris>—** fw-/T'(El(
is an isomorphism, as desired...
Moreover, by the quasi-coherence of crystalline
cohomology G.6), we may work over A = PCS," ) instead of
Then G.15) applies, so we need only check that R fy/c E
coherent on S...
Suppose that in the notation of G.17)
A is a complete discrete valuation ring (necessarily of
characteristic p and absolute ramification index _< p-i)
P = I = the maximal ideal of A...
In this chapter we shall study
this action, in particular, its relationship to the Hodge filtra-
filtration on crystalline cohomology (as determined from the ideal Jv/c
The main global applications are Mazur's theorem (8.26), which say
that (with suitable hypotheses on X) the action of Frob-snius de-
determines the Hodge filtration on H*R(X/k), and Katz's conjecture
(8.39), which says how the Hodge filtration limits the possible
"slopes" of Frobenius...
Neverthel
has a left derived functor, in a slightly extended sense,
ause of the following simple result:
9 Proposition...
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«)...
•'•The reader who so desires can now skip to Katz's conjecture, p
Here we shall develop only those properties of this structure
we need for the applications, and refer the reader to Mazur's
papers for more details...
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...
We have a corrmutative diagram of triangles:
К рГ J< РГ
•Ь(с
Ln
1- t-
О > EuAJ r+1 >IRuftJ Г >П 'и
Moreover, M is'contained in the image of the injective
so that МГс—*H*ris(X/S) factors through ar...
(In particular, the endpoir.t of the first lies to the left of
the endpoint of the second.)
Proof...
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 ...
Roby [1,2] in which the many interesting proper-
properties of the functor гд(м> аге developed...
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.....
Since the complex F.fiM is flasque, there is an exac
sequence:
0 -> R-'-lira H1~1(F.8M) -* H^-Clim F.8M) ¦¦ lim H^F.SM) -*¦ 0 ,
and hence HX(F 8M) = 0 for i < m ...
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