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Berthelot P., Ogus A. Notes on crystalline cohomology (Princeton, 1978)(T)(264s)_MAh_.djvu |
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is isomorphic to the PD polynomial algebra 0у<5т-.-Ч >» by C.3
Thus, Ь(Пууд)т is just the deRham complex of the PD polynomial
algebra 0™.<?,..-? >, relative to T...
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...
In order to study the base changing properties of the
cohomology of a crystal, we first need a fancy form of the ad-
adjunction formula in the derived category...
Using the adjointness of f* and fA , we obtain
natural map:
Adf: Ж НоШд,A^*Ё* ,F') = Нотд, (f*L',Г ) =« НотдA/, f*I*)—>
> HomA(L* ,J')*-^-HomA(E' ,J*) = Ж Нотд(Е# .Kf^F") ...
We begin with a very special case, assuming that X
lifts to a smooth affine scheme Y over S...
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
п п-1
Tensoring with the complex Kfy.^ (E) in the derived catego
and using the base change isomorphisms
X
for v - n and n-1, we get a triangle:
Pfx/s* E *fx/s*E
1 n
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...
an
verse system of abelian sheaves on X, snd suppose that
exists a basis of open sets of X satisfying:
(a) Hq(V,Fn) =0 if q > 0
(b) The inverse system H°(V,Fn) satisfies the Mittag-
Leffler condition (ML)...
We had best begin by remarking that we have a
language in the second statement, since we have written IR
in a derived categroy...
Indeed, it suffices to check G.23.1) after restricting to
¦(X/S ) for all n, so that it follows from E.26) and C5.27)...
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 Jy/s^'
The main global applications are Mazur's theorem (8.26), which says
that (with suitable hypotheses on X) the action of Frobenius de-
determines the Hodge filtration on H* (X/k) , and Katz's conjecture
(8.39), which says how the Hodge filtration limits the possible
"slopes" of Frobenius...
Indeed, since we
I/O I/O
are working with p-adic formal schemes, it is enough to check
this mod p, and we have Wv /Q о Fv ,<, = Fv and
*0/b0 0 0 x0
Fv /o ° Wv .„ = Fvt ...
Ц We shall be especially interested in the case where A
is the abelian category of complexes of an abelian category...
of this complex is
It is clear that these sheaves satisfy the hypotheses of G.20)
hence are acyclic for it..: МхХ +Х ...
Since TD/S is a
morphism of complexes, (8.21.1) follows, and (8.21.2) is ал
immediate consequences...
In the derived category, this trans
lates into a triangle:
]Ln/n'
Ln1 > In
where ILn/n' is the mapping cone of Ln' ¦* Xn ...
It is nonetheless apparent that there should be some r
tion between the Hodge filtration and Fv/C, and from (8.23
л/ о
we can also expect the conjugate filtration F" (associat
the spectral sequence Epq = Hp(X,Hq(n^/s ))->ffli(X,U^
play a role...
Now this spectral sequen
is, after renumbering, the spectral sequence of the canonical
filtration [1,1.4] T...
To prove the theorem, and perhaps to give some
insight into its meaning, it is helpful to baldly list the
inequalities which it asserts...
The reader may wish to note that we have in fact proved a
slightly stronger inequality than claimed, which he can work
out according to his needs...
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