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Berthelot P., Ogus A. Notes on crystalline cohomology (Princeton, 1978)(T)(264s)_MAh_.djvu |
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'¦ * *
arrow jy*LE) -*• (Xo/) E a y*(X*(E))...
Suppose S is quasi-compact, f: X + (S,I,Y) is
quasi-compact and quasi-separated, and fV/o: (X/S) ...
If E' and F" are two such complexes, the
natural map L'(E') IB L* (F-) -* L'(E*®F') is hence a quasi-
isomorphism, and since L" is also compatible with shifts, one
see that if С(u) is the mapping cone of a morphism of com-
complexes u: E' -¦ F', then there is a natural quasi-isomorphism
C(L'Cu)) ->L'CCCu))...
Then as in the proof of the previous re
if K* ~is"~acyclic in degrees < m, one has a quasi-isomorph
P- -> A', with' АШ = p'VdCp), An = 0 if n < m, and An
if n > m...
p
To prove Proposition G.19), note simply that if
Ee (X/sA)cris ' then C*E)(n,(U,T,6)) aE(U,T,<5)- for a11
n >> 0, so G.21.2) is certainly satisfied, and
(i*3*E)(U)TN) = ii?<i*E)(n>(u>Tj6)) = E(u,T,e)'- D
n
Because of their functoriality, the constructions of
(X/S) ...
Suppose X/SQ is smooth, quasi-compact, and
quasi-separated, and E is a locally free, finitely generated
crystal of 0x/g-modules in (X/S) ...
Indeed, since we
I/O I/O
are working with p-adic formal schemes, it is enough to check
this mod p, and we have Wv /Q о Fv ,<, = Fv and
*0/b0 0 0 x0
Fv /o ° Wv .„ = Fvt ...
If e: Z •*• Ш is a function and if K'
a complex in A , then "K"" denotes the subcomplex of K' give
by:
,.i e(i)vi о j-lr Ei(i+l)vi+lT
K?=p К П d [p К j...
Consider the topos (X/S)cr^s described in G.17) (recall
that we are writing S for S), and regard the PD filtration
JjJs as a filtration on the object 0x/s (we set j^j = 0X/S if
i <...
This translates into a statement in the
derived category: There is a commutative diagram (not a triangl
,y
We call this the "shifting diagram"...
•*• K'9Z/pZ are isomorphisms for г ¦>> 0, and assump-
assumption (a) implies that TH^K') + ЗН'Чк'
is surjective for
all i ...
Since taking cohomology commutes with flat base change
it follows that js 8id_ is an isomorphism for all s,t,r ...
To prove the theorem, and perhaps to give some
insight into its meaning, it is helpful to baldly list the
inequalities which it asserts...
It is a subgroup of
the group of units of R[[T]], and an R-^module using the rule
(rf)(T) = f(rT)...
To see that $ is a PD morphism, it suffices
check elements of the form x , since these generated Г (М
A
as an ideal...
Moreover, since DQ e Dft^D^' we can cnoose a complex E,* of
finitely generated protective A^-modules representing !>c, and if
BQ is a noetherian A.-algebra, Die...
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