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Berthelot P., Ogus A. Примечания относительно прозрачной когомологии (Принстон, 1978) MAh

Berthelot P., Ogus A. Notes on crystalline cohomology (Princeton, 1978)(T)(264s)_MAh_.djvu

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Date Jun 22, 2005

Cites: Using E.25.1) we see
that the source of this hoped for arrow assigns to any
u: T + Dx y(Y) the sheaf L(E)T> and by E.26) its target assig
to u the sheaf u*X*(E)...
Suppose S is quasi-compact, f: X + (S,I,Y) is
quasi-compact and quasi-separated, and fV/o: (X/S) ...
(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...
is defined by a sub PD ideal, Dx y(Y) = Y, and the crystal E
corresponds to an fly-module E = Z- ,v...
Because Hf^^^(E) is bounded above, we can find a complex
L' of flat Og-modules representing it, still bounded above...
Note that if T 6 Cris(X/S ), the sheaf
i? (T) is representable, in fact by TKg S^ (exercise)...
In the second part V)e interpret the calculations in terms
of crystalline cohomology and obtain C.20), which does have
global meaning...
Notice that if F is decreasing, we may as well assume
that e is also decreasing (= nonincreasing) because F A = FA,
where V is the maximal decreasing function less than e , given
by U(i) = min{e(i'): i1 i...
It follows from
G.22.2) that these sheaves are acyclic for v .„ , so that:
But G.22.1) tells us that У.* is also exact, so the qth
•'•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...
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)...
in the lemmas which follow, v.-e assume only that >;/sQ i&
and smooth, and state the additional hypotheses as we nee
If X does satisfy all the hypotheses of the theore
base changing theorem for crystalline cohomology and the
ness assumption show that H*rig(X/SHo 0s =sH*R(X/SQ), an
3 0
particular, the latter is locally free...
Now this spectral sequen
is, after renumbering, the spectral sequence of the canonical
filtration [1,1.4] T...
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 Identify the Image of H1(X'/S,^js) as Me, It clearly
prove that H " $ (Me ), In the notation of (Й-.24)...
Expanding in powers of T, we see that ther
are unique elements D (p)D(z) of N8R such that
S (p)-(z) = j/l](p).(z)f ...
Finally, it is clea
J- у I1 ¦ •¦ Л* Лш J1 Л,
that iji is unique, because У(М) = Г (M) generates Г.(М) а
an A-PD-algebra...
In fact, one sees easily by descending in-
induction that the complex K^_1 splits: K^-l = Pn-1 6 Pn-1 '
with the obvious boundary maps...
If, moreover Nn has
finitely generated cohomology, it is isomorphic to a complex
M of finitely generated A.-modules, and since each term of
the complex F.6M will involve a uniformly finite direct
sum of terms as in СБ2.2), we see that the arrow is an
This H* is again a complex of finitely
generated flat A-modules, which evidently satisfies the hypo
theses of (Б.9)...

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