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Berthelot P., Ogus A. Notes on crystalline cohomology (Princeton, 1978)(T)(264s)_MAh_.djvu |
Size 2.0Mb Date Jun 22, 2005 |
rD л ,Y s >
(jy (Xoy)*t)T =* PT*Py*(H), where py: D^CTx*) - Y is the
natural projection...
) which sends Fmn^ <?> to J^ , and a
chain homotopy s: ion - id^ , (the 0™-linear map sending
С d? to $ -* ) under which f^ <?> is invariant...
D
In particular, E = <iCrig E)y = X*(ED), with D = »ХЛ(У)
and icris (E)_is the crystal on Y/S defined by E viewed as an
HPD stratified 0,,-module...
Finally, if K* is bounded above and N" is bounded below,
I? НоШдСМ*,N"] is defined to be the class of the complex
Нотд[М",1'], where I* is an injective resolution of N*...
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...
There is even a morphism
?71 ' With
uv/c ft(F)(n,U) = »r(u/S ,j*F), and a conunutative diagram:
л/Ь« п п
<X/Vcris
ux/s
X/S...
Even though we have compatible isomorphisms
BuX/S*Jx7s En * Fm(ETiefiD /S } for each n ' this doesn>t
give us the theorem, because one cannot work locally in t
rived category...
It seems
reasonable to attempt to define the derived functors of the
s-construction, in the context of the filtered derived category...
there is a commutative diagram as shown, in which У is an
E
x/s
'FX/S*KuX/S*°X/S
This diagram is functorial in X and e , and agrees with the
diagram (8.8) in a lifted situation...
I do not know much about
its meaning — perhaps it is correct to think of it as a p-adic
elaboration of the "conjugate filtration" of hypercohomology...
(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...
i + j=n
Til
Let ГД(М) = G,<.H)/1 .(M), and for each x 6 M, let x
denote the image of (x,n) in r.Oi)...
Using the fact that formation of Г commutes with base
•- n
change, we see that x—i-x is a polynomial function of
weight n...
This is a consequence of th
definition, because in ГД(М) we have, from the defining re
tions cf type III, xx ¦ = (n+l)x , and hence by indue
xn = n.'xCn]...
= D as in (B2.1); then Dm@ An= Fm9^ An ,
Tii m
and since the system is flasque, ]Rlim D 9 A = lini(F 9...
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