Dr. Katz DEq Homework Solutions 32

Dr. Katz DEq Homework Solutions 32 - 32 N.M. Katz: is...

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Unformatted text preview: 32 N.M. Katz: is defined (because, in dimension n, all the "lower" ones H-W(/), i <a, h ave source and target the zero module, hence are isomorphisms). In f act, it is none other than the composite (analogous to (2.3.4.1.3)) 1s,n--a,a cona.~2 ~.~l~n f [r ~ Jt~, j , t'~X/S) a,n--a ~ E1 (2.3.6.2) F ~ ( R " - " f , (f~x/s)) .................................. .+ R"-"f,(~x/s). n-w(.) :: (2.3.6.3) Proposition (Igusa, Manin). A ssumptions as in (2.3.6), s uppose S affine and so small that all the Hodge cohomology sheaves are free (9sm odules. Let co1..... me be a base of R ~- af , (~ax/s) ' a nd denote by (aij) the m atrix in Me(F(S, (_gs))of H -W(a) w ith respect to this base: (2.3.6.4) H-W(a) (F~s (coi)) = Z a ji o)j. i (2.3.6.5) C onsider the dual basis o9",...,o9" of R N+a-nf~(~'~N~sa ). B y d uality and the definition of the integer a, it is also the least integer with R N+"-"f, (12~7s~) non-zero, so that the degeneration of the Hodge =~ De R ham spectral sequence gives an inclusion (2.3.6.6) R " +~ "f, ~ (a;,/s). (2.3.6.7) L et ~1 . .... ~ e be T-linear differential operators on S which are in the algebra generated by Der(S/T), which operate on the De Rham c ohomology sheaves Ri f , (Ox/s) via the Gauss-Manin connection 17.Suppose ~ t hat m* .... ,co*, considered as sections of R 2N-n: t ~*x/s1, satisfy the J,~ d ifferential equation (2.3.6.8) Z V(~j)(~o*)=0 in Rzu-"f,(~2]/s). J T hen each column of the matrix (aij) satisfies the differential equation (2.3.6.9) Z ~j(aji)=0, J for i=1 ..... (. P roof T he proof is based on the fact that the composite p-linear m apping (2.3.6.9bis) R "-"f,(fPx/s) F'~b. , F , s ( R , - , f , (fPx/s)) n'W(")~ R "-~ f , (fPx/s) r m ay be factored through the subsheaf R , J ,~q2o ~v of h orizontal s ections x/s) (el. (2.3.1.3)), as expressed in the commutative diagram ...
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This note was uploaded on 12/21/2011 for the course MAP 4341 taught by Professor Normankatz during the Fall '11 term at UNF.

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