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Dr. Katz DEq Homework Solutions 45

# Dr. Katz DEq Homework Solutions 45 - iq(3.4.1.5 1 b I(D(io...

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Algebraic Solutions of Differential Equations 45 We next define a (not necessarily integrable) T-connection 17 on the simple complex deduced by "totalization" from the Cech bicomplex, which yields the Gauss-Manin connection upon passage to homology (cf. [24, 35]). For any DeDer(S/T), we denote (3.4.1.1) D(i)=the unique element of Dero(VjT ) which extends D and which annihilates the chosen local coordinates x~(i) .... , x,(i), and by (3.4.1.2) Lie(D(/)): f2~, s(log D)~ f2~,/s(log D) the "Lie derivative with respect to Di". For each pair of integers i<j, we denote by ~ o--1 (3.4.1,3) I(D(i)- D(j)): f2v,~vj/s(log D) fJv, nvj/s(log D) the operator" interior product with the S-derivation D(i) - D(j)" (cf. (1.2.2)). The connection 17 on C'({Vi},g2]/s(logD)) is given as follows. For a fixed integer b, a section (3.4.l.4) o)~ Z C"({V~}, ~b/s(log D)) a and a simplex io <"' < iq, (V (D)(o)))(io, ..., iq)= Lie(D(io))(oo(io
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Unformatted text preview: .... iq)) (3.4.1.5) + (- 1) b I(D (io)- D(ix))(og(il . .... iq)). Thus 17(D) is the sum of two terms. The first, of bidegree (0, 0), is the cup-product with the 0-chain {Lie(D(/))}. The second, of bidegree (1, -1), is the cup-product with the 1-cocycle I(D (i)-D(j)). This explicit construction makes clear computationally the truth of (1.4.1.6), (1.4.1.7), and (2.3.0.1). In order to prove 3.2, it suffices by linearity to establish the com- mutativity of the outermost square of (3.2.0), and after "contraction" with any D~Der(S/T). Thus we must prove commutative the diagram a b R f,(JY (f2~/s(logD))) r ~R,+lf,(gfb_l(f2~/s(logD))) (3.4.1.6) l~e-l~ l~e-I ~ R"f,(ab/s(log O)) (-1)b+10(D), RO+l f,(f~xSs~(log D)). Let us explicate the arrows in this diagram. The vertical ones are deduced from the morphisms of sheaves (3.4.1.7) f~bx/s (log D) "*, f~x,,,/s (log D ~"))--~21 , o~ b (a}/s (log D))....
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