Algebraic Solutions of Differential Equations
21
is endowed with the GaussManin connection (cf. [24], 3.5). For reasons
which will soon become apparent (cf. (2.3.3)below), we call this the
conjugate
spectral
sequence,
and
the
corresponding
filtration
of
R"f,(f2~/s(log D)) the conjugate filtration, noted
i
F~'on. This filtration is
horizontal,
and is every bit as interesting as the Hodge filtration.
(2.3.1)
Let us explicate the
Ez
term of the conjugate spectral sequence
with the aid of Cartier's isomorphism (2.1.1).
(2.3.1.1)
Lemma. The
Cartier isomorphism ~ defines an isomorphism of
(gsmodules for each pair a, b of nonnegative integers
(2.3.1.1.1)
Ea,~_
~
b
co, 2  R f, (Jr~ (f2x/s(log D)))
,
R"f~,P)(tr*(Qbxls(log D))).
Proof.
From the commutativity of the diagram
X
r
~ X~p)
S
and the fact that F is a homeomorphism, we have isomorphisms
(2.3.1.1.2)
~o,E3
'b =
Raf, (~'r ~b (t2~,./s
(log D))) ~
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 Fall '11
 NormanKatz
 Logarithm, Isomorphism, Group isomorphism, Cartier, Cartier isomorphism

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