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Size 2.0Mb Date Jun 22, 2005 |
Oy-modules and HPD differential operators to the category of
HPD stratified Oy-moduies and 0v-linear horizontal maps...
Then the map
Ly(d): Ру/5A)8Пу/5 * pY/SA)enY/S behaves as follows:
Ck,3 [k ] n [k.] [k.-l] [k* ]
Z 5 вш) = a I 51 L ...5i г ...?п n
[к,] [к]
Proof...
Let К denote the PD filtration C.2Ц); it follows from the
local formula F.11) that L(d) maps K^'^HU^/S) into
Set F^L(nqy/s) = (С[га'Ч] L<.^/s>, so that
is a subcomplex of L(fJy.g)...
?
Using the same technique we can also obtain resolutions of
other crystals of 0^ .g-modules...
How E^ is the cohomology of the oompl
E0 = fX/S<,i(p)crisJ(p)cris(E') and since the comPlex 3(p)eris(
is a resolution of j, ...
E = E1 and we obtain (from the identity map idPI) a
СП S t*
morphism in the derived category: E -> К6СГ1ЗАЕ'-• Applying th
functor ^x/S » we obta^n an element of
4 ,Kfx/s Kgcr.BAE') * Ж HomgORf^E ,KuJRfx,/s,ftE
By the adjunction formula for u, this in turn is isomorphic to
К Horn (Lu*Rfx,s E , Kf^, ,g,AE'), which is where we wanted to be...
First suppose that M is quasi-coherent
and consider the Og-algebra О-вМ , S1 = 5peCg( 0gffiM) .• The ideal I1
generated by M and I in 0_, has a PD structure У , and there
is a PD morphism u: (S',i',y') ¦* (S,I,Y)...
For each n, we let An = A/Pn+ , S = Spe.a An „
S = Spe.c A, and § = 5p(J A (for the P-adic topology)...
Notice also that the above diagram makes
Y0/S0 Y0/o0 Y0
sense for any Y/S, not necessarily smooth...
(In the presence of torsion, this may only define a preshea
and K^ is the associated sheaf.)
It is clear that a morphism f: A" •* B" induces a morphism
fe: Ae*Be for a11 E ' and tnat this defines a functor...
Consequently, we
deduce an isomorphism:
Now recall that the qth term of
Ip?(i)FJ.LD/b_) : ?p?(i)j^<J[i-*Vn§/s) ...
Indeed, if Rq: A4 + Bq-1 is the homotopy, it follows immediatel
from the definitions and the fact that r\ is increasing that r5
s A^ into B^~ , and hence induces a homotopy R^: f
course, formation of K' is not compatible with translat
ce doesn't preserve triangles, nor is it exact...
That is, there is a c'ommutat
diagram:
FX'fiDY1 V(Z)/S *FX/S*(ni>y Y(Z)/S)n
A ,1 Л,I
FX'nDx, Y(Y)/S * XjY
Moreover, if (Y.F.,) and (Z,F2) are any two lifted situations
which X embeds, then we can also embed X in (YxZ, Fy x F^)
(as a locally closed subscheme, but no matter) — and this
both to (Y,FV) and to B.,F«)...
Now this spectral sequen
is, after renumbering, the spectral sequence of the canonical
filtration [1,1.4] T...
If x € Im(*) Л рг H^ris(X/S), then x = рГу for
some у such that nv(y) 6 ТХ~Т H^(X/S0)...
In particular, linear maps define polynomial functions
of weight one, and if -h:N -> N1 is linear and f 6 P (M,N),
then hof e Pn(M,N')...
Find a surjective map P -+• A such that P -»• PBQ is
tive, and let I be the kernel of this map...
In fact, one sees easily by descending in-
induction that the complex K^_1 splits: K^-l = Pn-1 6 Pn-1 '
with the obvious boundary maps...
Then the flasque
complex S(L') represents LS(M), hence lim S(L') represents
Д? jLim Ь S(L") а К lim D =s M ...
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