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Berthelot P., Ogus A. Notes on crystalline cohomology (Princeton, 1978)(T)(264s)_MAh_.djvu |
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sending x to хв1, which is compatible with HPD stratifica-
stratifications...
First let us describe the ideal к?т"' in terms of
i
local coordinates, where T = (U,T,<5)...
Let E be a Vv Y(Y)-module with HPD stratifi-
cation, let E be the crystal on X obtained from E by F.6),
and let ^®n^T) (Y)/S ^e *^e comP-'-ex °^ sheaves on X ob-
X,Y
tained from the connection on E ...
This holds for any
morphism of ringed topoi, as explained in [SGAUXVII], where it
is called "trivial duality"...
For each n, we let An = A/Pn+ , S = Spe.a An „
S = Spe.c A, and § = 5p(J A (for the P-adic topology)...
We sha
say that a sheaf of ^v/q -modules is a "crystal" iff the map
( T) "* F(n'' T1) are isomorphisms f°r every u...
We had best begin by remarking that we have a
language in the second statement, since we have written IR
in a derived categroy...
Suppose W is the Witt ring of a perfect field к of
characteristic p, and X is a smooth k-scheme...
Since the value of e(i) for negative i
has no effect, we may assume that e(i) = e@) - i for all i >...
If k6Z, let <k> = nin{ord (pr/r!): r>.k>,
so that (p)Ckj = (p<k>) ( = the unit ideal if к <...
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^)...
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«)...
(X/S)} onto
Fcon HDR(X/V •
(8.26.3) The diagram below commutes:
С
(The diagram makes sense because of Lemma 8.27, which tel
that the Hodge and conjugate spectral sequences degenerat
suitably.)
Proof...
(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...
Then if
t = bQ+--.+b , one sees from the above that A 2L В if
30+"-+8т^ V'+tm for a11 " *
8.37.2 If M'—>M is a span with Hodge numbers e ,ех.....
We regard D as an endomorphism of P(M,N)
л
and S = У D 1 T1 as a formal power series with coefficients
x i=0 x
the ring of all such endomorphisms...
(B.1.3) The derived functor of lim = ГШ, ) is defined
all of DOHjA.), not just on D QN,A.)...
Let D be its image in D(li,A.);
By (B6) D e D~CN,A.) and is consistent, hence
by (B.7), Klim D € Dft(A) • Jt i-s clear from (B.5...
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