Dr. Katz DEq Homework Solutions 7

Dr. Katz DEq Homework Solutions 7 - .., r), ds, (#=l, . ..,...

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Algebraic Solutions of Differential Equations 7 affine open sets q/i, and cover X by affine open sets V~ such that (1.0.3.1) each q/= q/i is 6tale over At (r depending on i) via local coordinates sl, . .., s, each V= V i is 6tale over At, (n depending on i) via local coordinates Xl, . .., x, the branches of D which meet V~ are defined by the equation x, = 0, v = 1 .... , c~ (a depending on i). Then, over, V the sheaf Dero(X/T) is a free (~v-module with basis (1.0.3.2) x~ -- (v = 1, ~), (j = ct + 1, n), a ~x~ ..., ~-~f .... ~ (#=l, . . . , r) and Der o (X/S) is a free (~v-module with basis (1.0.3.3) x~-- (v = 1, ct), ax, ""' 0xj (j=0~+ 1 ..... n). Thus, over V, f2~/T(1Og D) is free on (~v with basis (1.0.3.4) dx~ (v=l,. ..,~), dxj
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Unformatted text preview: .., r), ds, (#=l, . .., r) Xv while f21/s(log D) is free on (_9 v with basis dxv (1.0.3.5) (v= 1, . .., c0, dxj (j=~+ 1, . .., n). Xv (1.0.3.6) Clearly the formation of f2]/s (log D) commutes with arbitrary change of base S' -~ S. (1.0.3.7) Remark. (When S is of characteristic zero, the complex I2]/s(log D) is quasi-isomorphic to j. t2~:/s. This is not the case in general, because j. f2g./s has "too much" cohomology, while I2]/s(log D) has only "geometrically meaningful" cohomology, whence our "preference" for t2]/s (log O).) 1.1. The Kodaira-Spencer Class From the definitions, it follows that we have an exact sequence of locally free sheaves on X (1.1.1) 0 -~f* (f2~/r) --~ t2~/r (log D) -~ f2~c/s (log D) -~ 0...
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