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Berthelot P., Ogus A. Notes on crystalline cohomology (Princeton, 1978)(T)(264s)_MAh_.djvu |
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Size 2.0Mb Date Jun 22, 2005 |
the Ly construction by working it out in detail using local
coordinates...
First observe that there
is a functor L which associates to any A-module E a flat
A-module L (E) and an epimorphism L (E) -» E...
The key concept is that of a
perfect complex.' This notion is defined and studied in [S
for nonnoetherian ringed topoi...
Then the topos IN of sheaves on К is, just the cate-
category of inverse systems, and ГСЙ.А) = lim A ...
The Frobenius auto
morphism of W is a PD morphism, covered by the absolute Frobenius
endomorphism F....
;
then we can find a gauge c' > e such that e'(i) = e(i) for
almost all i and such that c1 is steep at j and j+1 whenever
0 <...
of this complex is
It is clear that these sheaves satisfy the hypotheses of G.20)
hence are acyclic for it..: МхХ +Х ...
Since TD/S is a
morphism of complexes, (8.21.1) follows, and (8.21.2) is ал
immediate consequences...
This translates into a statement in the
derived category: There is a commutative diagram (not a triangl
,y
We call this the "shifting diagram"...
°r-l °r
To prove this, observe that a ^ is a simple augmentat
of a at r , so we have commutative diagrams:
r-1
Н (K*eZ/pZ)[-r]
О >Tr_1(K*9Z/pZ) >Tr(K'eZ/p2) >Нг(К*в2/рЖ)[-г] >О
r-1
Hi(Tr_1)
¦ Hx~r(.HT)
First of all, because K* is bounded, the maps K* ¦+ K' and
Tr(K"8Z/pZ)...
Of
course, all.these are simple augmentations at some, r' >_ r, so
by case 3, the maps H*(K' ) + B*(K" ) are all injective...
Since the length of this is less than or equal to
h + h +•••+ hm" (by the Cartier isomorphism), the desired
inequality follows immediately from the induction assumption...
In
particular, the index of its first nonzero Hodge ;":ur;iber is clearly
if e + • • •+e-'~ <_i ? e°+...+e^ ...
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...
For example, let к be a
of characteristic p > 0, let A = k[T], and let 1= (T) ,
M a A/I, and К* г A/TPA...
An object D of D"(]N,A
has finite tor-dimension iff there exist a and b such tha
i I/
for all n and all A -modules M, H (Dl M) = 0 unless
n n л
¦n
i 6 [a,b]...
Now write P^_^ as a quotien
of a free An-module P^ and take K^ = P^ © P^ with the
obvious boundaries...
Suppose L* is a complex of flat A-modules s
a) The natural map L* -* lim L*0A is a quasi-isomo
b) Each H (gr.L*) is a finitely generated gr-A-mo
Then:
B.8.1 The inverse systems H1(L'8A ) satisfy M.L...
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