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Dr. Katz DEq Homework Solutions 46

# Dr. Katz DEq Homework Solutions 46 - {I(D(i-O(j The upper...

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46 N.M. Katz: In order to render the vertical arrows still more concrete, we introduce, for each open set Vi, a p-linear endomorphism O5 of ~v,/s (log D), whose image lies in the subsheaf of closed forms, and which "lifts" the mapping 3.4.1.7. Indeed, the formulas (2.1.2.1) give such a lifting: we require that o5(1)=1 (dx~(i) ~ = dxv(i ) O5 \ xv(i) ] x~(i) for i= 1 ..... os(dx~(i))=(x~(i)) p-~ dx~(i) for i=~+ 1 ..... n (3.4.1.8) os(co ^ ~) = os(co) ^ O5(~) os(co + ~) = os(co) + O5(~) os(h co) = h p os(co) We then have our desired lifting (3.4.1.9) for h~(gv~. f~v,/s(logD) ~=' , closed forms ~v,/s(logD) tcanonical projection ~b'~o~ 1 D ~ ( v,/s( og )) (3.4.1.10) The lower horizontal arrow in (3.4.1.6) is, as previously (cf. (1.1.3)) noted, (- 1) b+1 times the cup-product with 1-cocycle
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Unformatted text preview: {I(D(i)-O(j))}. The upper horizontal arrow in (3.4.1.6) is slightly less straightforward to explicate. Let ~ be a section of Raf. (j~ffb (i2~/s (log D))). We may represent by an a-cochain ~ of closed forms (3.4.1.11) z~ca({v~},P~x/s(logD)), dz=O whose image in ca({ Vii}, ~b(~2]/s(log D))) is a cocycle representing ~. By the degeneration of the conjugate spectral sequence at Ez, we may choose z to be the component of bidegree (a, b) in a total a + b cocycle a which lies in (3.4.1.12) F~,= 2 ca+i( {Vi}' f~x~-s'(logD)). i=>O The section ~, (D) (~) of R a + i f, (ocgb- 1 (f2~/s (log D))) is then represented by the component of bidegree (a + 1, b- 1) in the total cocycle (3.4.1.13 ) ( V (D) p - V (D')) (a)....
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