Dr. Katz DEq Homework Solutions 9

Dr. Katz DEq Homework Solutions 9 - Algebraic Solutions of...

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Algebraic Solutions of Differential Equations 9 The associated graded module is given by (1.2.1.3) grj~ (A~174 Consider the exact sequence (1.2.1.4) 0 -~ gr~ -~ K~ 2 -, gr ~ -~ 0 whose term of degree v is (1.2.1.5) O-~ f~| ~-, K~ ~ff)-~ W ~-~O. (1.2.1.6) For each integer v> 1, we define in this way a functor W from the category EXT(~ fq) of extensions of ~ by f~ to the category EXT (A~ ~, f#| ~-a ~) of extensions of W o~ by fr | W-a ~ Passing to the (groups of) isomor- phism classes of objects of these categories, we obtain a morphism, still denoted A ~, (1.2.1.7) A~: Ext~x(~, fr Ext~,,(AV~,c.9| ~). Because ~ is locally free, the sheaf Ext~x(~, ~9) vanishes, and the local =~ global spectral sequence of Ext furnishes us with an isomorphism (1.2.1.8) Ext~s (~, fq) ~ H a (X, Homr f9)). Similarly, the local freeness of A ~ ~ furnishes an isomorphism (1.2.1.8bis) Ext~,,(A~, (r174 ~-~ ~),,~HX(H, Homr ~| ~)). Let us make explicit the two identifications (1.2.1.8) and (1.2.1.8 bis),
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