Dr. Katz DEq Homework Solutions 35

Dr. Katz DEq Homework Solutions 35 - ",r,ex~0 it...

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Algebraic Solutions of Differential Equations 35 from which it follows that n, (f~sl| -*) is a free tgs-module, with basis the differentials X w dxl dx,+t A "'" A --, Y', Wi=d, Wi>0 for all i. (2.3.7.9) H xl x,+l The image of these differentials under the residue isomorphism (2.3.7.6) are the differentials on X" X w dx I dx. A'"A --, E W/=d, o)(W) = c3H xl x. X,+t 8X,+1 W~>0 (2.3.7.10) x w'-I . ..xW${'-XdXl A . .. A dx, m Oh Ox,+l for all i The bases m(W) of R"f,((_gx) (2.3.7.4) and og(W) of f,(f2~/s ) are dual to each other under Serre Duality: W),og(V))=~I if V= W (2.3.7.11) (m(- if V,W. (2.3.7.12) Suppose once again that S is of characteristic p. Then from the commutative diagram of sheaves on p~+l 0__). ~p (__ g 0 H )(Qp , ~X _._+ 0 (2.3.7.13) l~p-,.r.b, lr.b. F.b. 0-,e,(-d) " ,r
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Unformatted text preview: " ,r ,ex~0 it follows that the p-linear endomorphism of R"f, ((gx) induced by the p-th power mapping (gx---~ d~ x corresponds via the coboundary iso- morphism (2.3.7.2) to the p-linear endomorphism of R" + 17z, (d~p (- d)) induced by the composite (gp(-d) --P'thp0we, ~e(-pd) HP-' ,~p(__t0. This permits the calculation of the Hasse-Witt matrix of a hypersurface: (2.3.7.14) Algorithm. Assumptions as in (2.3.7.0) and (2.3.7.12), the Hasse- Witt matrix in dimension n of a smooth hypersurface X cP~ +1 defined by an equation H=H (X1 . ... , Xn+ E)~F(S , (.gs)[X 1 . .... Xn+2] may be com- puted with respect to the basis {m(-W)[~.Wi=d, W/>0 for aU i} of R" f, ((gx) as follows: Raise H to p- 1-st power, and write it explicitly as a sum of monomials (2.3.7.15) H p- 1 =}-" AvX v. 3*...
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