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Berthelot P., Ogus A. Notes on crystalline cohomology (Princeton, 1978)(T)(264s)_MAh_.djvu |
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Since d is a differential operator of order one, we have a
diagram:
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 ...
Now if E" is a complex of A-modules, L*(E") is a double
complex of flat A-modules, and the associated simple complex
L'(E") is easily seen to be naturally quasi-isomorphic to E if
E" is bounded above...
(The fact that the arrow in the second line is a quasi-
isomorphism depends on the injectivity of the modules j .)
This gives us the desired arrow...
Of course, this arrow is the same as our fancy
looking Ь^е changing arrow, and the theorem is proved in the
special case...
Moreover, the functor' Гн F is
the inverse image functor of a morphism of topoi
^n: (X/Sn5cris "* (X/St)cris ' Finally> there exists a morph
of topoi j: (X/S.)crig -+ (X/S)cris , such that (j*(E)>n= i*
i.e...
We had best begin by remarking that we have a
language in the second statement, since we have written IR
in a derived categroy...
For each n, there is a natural base-changing
isomorphism:
X
Kr(X/S,E) в An >Kr(X/Sn,i*E ) •...
Let (А,1,у) be a P-adic
base G.17), and use the notations of G,17) except write S= Ssj{ A
instead of S = Spec...
* n
It is apparent from the explicit bases we gave for PD
envelopes C,32ff) that P is a flat 0O -module, so that the
П о
n+1
kernel of Pn+k * l?n is exactly p Pn+k ...
О
The above remarks provide us with a crystalline interpreta-
interpretation of the source of the arrow 4 of (8.8)...
We have a commutative diagram':
г.) 'H^.s(X/W)
Theorem (8.20) implies that the image of у in M is exactly M'
and the diagram implies that the image of 3H1(X, 3L<; +in) in M1
is contained in p M Л M1 , Thus, we get an induced map from
the image of q to M'/p 'M Л M , necessarily surjective...
But If r * 1 , gf (i) - 1 +c (r) is r +c (r) » r] (r)
equality if r-1, and If г г 1, gr(i) » r +e ( r) » r\ (г) г т\ A) ...
If R is an A-algebra and f G R[[T]], we say
that f is "of exponential type" iff f@) = 1 and
fCT^ + Tj) = f(T1)f(T2), for indeterminates T^ and Tj...
But this follows immediately from the
definition of S and the fact that Х.хвТ = хв\Т...
q
Our aim is to compare the categories D(U,A.) and D(A)
We have defined a functor F.lim: D(TJ,A.) -*D(A), and there
an obvious functor back...
B) follows, and than
one concludes easily that H1ORlim D) is тг-adicaily separated
and complete...
This filtrat
* n m n
is compatible with the J-adic filtration, and gr_H (L*) is
finitely generated grTA-module...
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