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Berthelot P., Ogus A. Примечания относительно прозрачной когомологии (Принстон, 1978) MAh

Berthelot P., Ogus A. Notes on crystalline cohomology (Princeton, 1978)(T)(264s)_MAh_.djvu

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Date Jun 22, 2005

Cites: There is a functor Ly from the category
Oy-modules and HPD differential operators to the category of
HPD stratified Oy-moduies and 0v-linear horizontal maps...
This ideal is J8Py/,g(l) + (О^вК), where
J С 0T is the ideal of 0^ and К С PY/g(l) is the ideal of Y -
an augmentation ideal...
There is a natural map: 0y -> PY/SA) ~ |"У@У)
sending x to хв1, which is compatible with HPD stratifica-
stratifications...
There
is a section тг: П^ < -* 0^ , (the 0T-linear mapping sending
efk] to zero for к >...
Furthermore, we let Y denote the induced PD structure on I/Pn
Suppose that X/SQ is a scheme to which Y extends...
For any filtered object (A,F) of A and
any function e: Z ¦*¦ M , we set: s
FeA = I p
iet
8.15il Remarks...
Recall that an
inverse system is said to be "eventually strict" iff the
transition maps are surjeetive for n >> 0...
Since Z has finite
projective dimension, this makes sense even if F* is unbound
we can take L"(F*) to be bounded below if F'is...
Indeed, the absolute Frobenius of P- lifts
to Pg , so X can be embedded in a lifted situation...
°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)...
A "span over W" is an injective W-linear map of free finite
rank W-modules of the same rank...
Form G = GA<"M), the polynomial
A-algebra on the set of indeterminates {(x,n):x e M, n ?K) ...
To prove that we get an isomorphism if M is protective
and of finite rank, we may assume that M is free, because both
sides are compatible with localization...
(B.I.4) A sheaf F on U is flasque iff the inverse syst
F is strict, i.e., iff all the maps F -+ F , are surjectiv
If F -» G is an epimorphism and F is flasque, so is G ...
Given F'_, ¦ Dn-1' find a pn e K~^An^ consisting of
protective A -modules and a surjective quasi-isomorphism
P" ¦* Ь'...
L
begin by recalling the following easy result, of which t
above is "the derived category version"...



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