This preview shows page 1. Sign up to view the full content.
48
N.M. Katz:
(3.4.3.0)
Assertion.
For any cochain te
C'({V/}, f~x/s(log D)),
the cochain
in C a+ a
({ V~}, ~x~ ~
(log
D)) which assigns to (io, .
.. , i,+ 2) the b 1 form
(1) b
~
Lie(D(io))kI(D(io)D(il))Lie(D(il))e(~i,(z(i~
.... ,i,+1)))
k+d=pi
(3.4.3.1)
( 1)bI(O(io)PO(ia)v)(~,(z(il
.... , i,+1)))
 (  1 )b +1 ~/o
(I (O (i0)  O (il)) (t
(is, .
.. ,
i,
+ 1)))
is a cocycle of closed forms, and is cohomologous to zero in
C ~ +1 ({ V/}, Jt~b' (O~/S (log O))).
(3.4.3.2)
In fact, we will prove that the cochain (3.4.3.1) is in fact a
cochain of
exact
forms, and so
vanishes
in C ~ + 1 ({
Vi }, ~b 1 (f2~/s
(log D))).
Notice that the occurence of ~o in the last line of (3.4.3.1) may be re
placed by ~, without modifying the class modulo
d(r (V~ o c~ .
.. c~ ~ .
... ~x?s 2
(log D)))
of the cochain (3.4.3.1), because both ~,o and 5/, restricted to V/o c~ V/, are
liftings
of c~loa* (3.4.1.7). This change made, the truth of (3.4.3.0),
and hence of 3.2, results from the following proposition (3.5.0), which
This is the end of the preview. Sign up
to
access the rest of the document.
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

Click to edit the document details