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58
N.M. Katz:
whose assosciated graded is given by
gr, w C~
V~}, t2~/s(log D))
(4.0.2.1)
=
~
C~
.... Di,/S)"
il<'"<in
The spectral sequence (4.0.1.3) is the spectral sequence of the filtered
complex of quasicoherent
(~smodules obtained from the filtered
bicomplex (4.0.2.0) by totalization.
The (not necessarily integrable)
Tconnection V on the Cech bicomplex C ~ ({ Vv}, I2~/s (log D)) constructed
in (3.4.1) preserves the filtration W. This provides a Tconnection on the
spectral sequence (4.0.1.5), which is that of GaussManin on the abutment.
To see that it is that of GaussManin on the E1 term, we simply note that
on the assosciated graded for W,
(4.0.2.2)
. C ~ ({ V~ n Dil n.
..
n
Di.}, ~2~)il
.....
Di,/S)
it induces the same Tconnection which the method of (3.4.1) would
construct, viewing the affine open sets Vv n
Dil n.
.. n Din
as a coordinatized
open covering of the smooth Sscheme Di~n.
..nDi, (the coordinates
being those on V~ whose restriction to Di~ n..n
Di,
do not vanish).
Of course, the truth of (4.0.2) could also be perceived by "pure
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This note was uploaded on 12/21/2011 for the course MAP 4341 taught by Professor Normankatz during the Fall '11 term at UNF.
 Fall '11
 NormanKatz

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