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Berthelot P., Ogus A. Notes on crystalline cohomology (Princeton, 1978)(T)(264s)_MAh_.djvu |
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be the crystal we get on Y/S by construction F.6), and L.'?
(or just L(E)) will be i* ...
There is a natural map: 0y -> PY/SA) ~ |"У@У)
sending x to хв1, which is compatible with HPD stratifica-
stratifications...
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...
(That is, L' preserves triangles.) We
now define Lf*(E') to be the class of f*L'(E") in the derived
category of A-modules...
On the other hand, E1 = g*E is a crystal on X1,
and :EfAt,eiE'is a bounded complex on S1, which is reasonable to
compare with Lu*B
In fact it is easy to obtain а шар (in the derived category
Lu*IR fx/s E +Kf',.g,E1, i.e...
If fx-/s = fx/s ° * ' etc" we Set
a commutative diagram:
Xu* *fx./s
Thus, it suffices to prove that the top arrow is an isomorphism...
The._reader may have already noticed an important consequence
the assertion G.8) that the arrow: Lu'OR^.g E "*:Rfx/Sft8cris* E
is a quasi-isomorphism: the target is a bounded complex by G.6),
hence H1CLu*Kfx/oftE) = 0 for almost all i ...
The category
(X/S.) ^s of sheaves on Cris(X/S.) can be interpreted as i
E.1): for each (n,T) € Cris(X/S.), a Zariski sheaf- F, T)
on T, plus "compatibility morphisms"
if
Notice that if n1 < n, we do not require the map
u~ F( T)*^"fn' T) to ^e an isomorph*sm> in general...
Thus, the
complex, as well as its cohomology, satisfies the Mittag-
Leffler condition, hence by [EGA Ojjj 13.2.3], we find that
Hq(lim r(V, Г)) = lim Hq(V,In) = 0 if q > 0,
i.e...
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...
Then Lemma (8.11) implies that 4? is a
quasi-isomorphism, and the proof of Theorem (8.8) is therefore
complete...
of this complex is
It is clear that these sheaves satisfy the hypotheses of G.20)
hence are acyclic for it..: МхХ +Х ...
Then
f(pnd)-n(i-l)a) _ db, (this makes sense because n is
increasing), and since H1(f) is injective,
pn(i)-n(i-Da=da, for ii
Then pr»(i)a = dpn(i-1)a', and p^1""*' ? aJ, so
Jpn(i)a] = 0 in H^A^)...
More-
Moreover, once the arrow * is defined, it "must be a quasi-
isomorphism, because this is a local question, and we may there
fore assume that (X,FV) lifts...
Now the map j? is just the obvious one:
Ht(Y,pe(s^y/s) ¦* Ht(Y,ny/s), and since ny/s is p-torsion free,
this map is an isomorphism onto pe s H (Yjfly.g)...
(X'/S) ¦»• H^ is^/s> deters
the (mod p) Hodge filtration of X'/SQ and conjugate fil
of X/?p, assuming the stated degeneracy and torsion hypo
Even without These hypotheses we car...
One deduces
immediately that the sequence
0 - lim(F.8M') -> lim(F.SM) -* lim(F.8M") -> 0
is exact...
Step 0: Suppose F" is a filtration on the A-module M wh
к v ^+
compatible with the J-adic filtration, i.e...
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