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Berthelot P., Ogus A. Notes on crystalline cohomology (Princeton, 1978)(T)(264s)_MAh_.djvu
Date Jun 22, 2005
Since d is a differential operator of order one, we have a
к 68idn к k+1
In this diagram» Lv(d) is the composition of the two horizontal
arrows, w: PV.<,A) ¦*¦ Py/o is the natural projection, and
: PY/S8nY/S + nY/S is the °Y"linearization °f d# N°W since
6 is a PD morphism, we compute:
/ n \
Recalling that w kills ? if I Jt I _> 2, we see that
n r j^_ i»l fir T
idnew9ido maps this to a • I ? x в?.8ш + a?L вш ...
Let К denote the PD filtration C.2Ц); it follows from the
local formula F.11) that L(d) maps K^'^HU^/S) into
Set F^L(nqy/s) = (С[га'Ч] L<.^/s>, so that
is a subcomplex of L(fJy.g)...
However, it is
important to notice that the process of embedding pieces of X
in smooth schemes is compatible with base change, so that G.6)
really gives an r which works not just for X/S, but also for
any X'/S1 obtained from base change via a PD morphism
(S',11 ,Y') - (S,I,Y)...
Of course, this arrow is the same as our fancy
looking Ь^е changing arrow, and the theorem is proved in the
By induction, IRf^.g E
has coherent cohomology, and the theorem follows from the long
exact sequence of the cohomology of a triangle...
It remains only to explain the last statemen
But H^ris(X) - lim H1(X/Sn ,0X/S ) =s lim H1(Y/Sn ,
by G.Ц), which is turn is the same as H1(Y/S ,ft* ...
Then Lemma (8.11) implies that 4? is a
quasi-isomorphism, and the proof of Theorem (8.8) is therefore
Assuming that H (f) is an isomorphism:
i) H1(f ) is injective: Any class in H1(AJ) is repre-
represented by some p a, with da =0, since A1
is torsion free...
That is, there is a c'ommutat
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«)...
formation of T preserves homotopies and quasi-isomorphisms,
hence passes over to the derived category, and pK* n K* = K' -
Thus, we have a canonical triangle:
8.23.6 Suppose that n' is a simple augmentation of n at j ...
Amazingly, Mazur's theorem asserts that
determines the (mod p) Hodge and conjugate spectral sequenc
(with suitable hypotheses on X)...
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...
exp(T)-= 1 + T/1+ T2/2!+-• • e exp(Q), there is a Z-ir.odule
M -• exp(Q) sending x to exp(T) , and hence a Z-algobrft
map а:Г^(К) -> Q sending mx to iri/n: e 5} ...
Cohomologie cristalline des schemas de caract?ristiqu
p > 0 " Lecture Notes in Mathematics № UlO> Springe
"K-theory and crystalline cohomology" to appear in
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