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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  Wa ~
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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This note was uploaded on 12/21/2011 for the course MAP 4341 taught by Professor Normankatz during the Fall '11 term at UNF.
 Fall '11
 NormanKatz

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