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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 |
then i can be identified with:
n
Т
All that remains is to observe that the projection id8w and
the homotopy ides are compatible with the filtration F
Since this is immediate, the proof follows...
= ^^zar ° K"
Let us begin by proving G.1.2) in the special case in which E
arises from an A,-module F with HPD stratification, i.e...
Because the isomorphism of G.3) is natural, we
can use it to compute the crystalline cohomology of maps, as
well as spaces...
Then bf*E' is f*L
К Hom^tOLf*E#,F") is Нотд,(f*L#,1*), and Ж Нотд(Е',lRfftF') i
Horn'(E*,J°)...
We are now ready to study the base changing properties of The
cohomology of a flat crystal...
The key concept is that of a
perfect complex.' This notion is defined and studied in [S
for nonnoetherian ringed topoi...
To define the cohomology we want, we construct a site
Cris(X/'S) which computes the limit automatically...
In this chapter we shall study
this action, in particular, its relationship to the Hodge filtra-
filtration on crystalline cohomology (as determined from the ideal Jy/s^'
The main global applications are Mazur's theorem (8.26), which says
that (with suitable hypotheses on X) the action of Frobenius de-
determines the Hodge filtration on H* (X/k) , and Katz's conjecture
(8.39), which says how the Hodge filtration limits the possible
"slopes" of Frobenius...
Let me now explain a special case of the main local result,
which gives a precise description of the image of
FY/S:
FY/S*
kiUs
ii/o ...
It is clear that what
we must prove j.s that the natural map:
-*lim Ep Ji: J is an isomorphism...
It follows that the arrow
V *V/S«JX/S * FX/S* ^"x/sA/S in the derived category
that we have defined is independent of the choice of embedding...
Fix an integer к > 0, and consider the follow
three statements:
<ak) The map K*(X,Fk^/s ) + Fk]H*(X,n^/s ) is an
isomorphism...
(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...
The Newton polygon of an F-crystal lies on or above
its Hodge polygon, and both have the same endpoint...
Before doing
so, it is convenient to investigate three interesting mapping
properties of the functor Гд...
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