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Dr. Katz DEq Homework Solutions 48

# Dr. Katz DEq Homework Solutions 48 - 48 N.M Katz(3.4.3.0...

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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=p--i (3.4.3.1) --(-- 1)bI(O(io)P--O(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
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