Algebraic Solutions of Differential Equations
31
are the usual perfect pairings of Serre duality, and the lower row of
(2.3.5.1.7) is simply the inverse image by Fab s of the lower row of (2.3.5.1.6).
(2.3.5.2)
Proposition.
Hypotheses as in
(2.3.5.1),
fix integers n>O and i.
The morphism
(2.3.5.2.1)
h(i):
Fc"o~i(R"f,(~'x/s))~R"f,(f2"x/s)/F i+j
is an isomorphism if and only if the morphism
(2.3.5.2.2)
h(Ni
1):
FcNo+
I +i"(RZN"f,(I?}/s)), R2u"f,(O}/s)/FNi
is an isomorphism.
Proof.
Just as in the proof of (2.3.4.1.7), we are immediately reduced
to the case in which S is the spectrum of a field K. By the autoduality of
the Hodge and conjugate filtrations (2.3.5.1.45), the
dual
of (2.3.5.2.1) is
the natural mapping
(2.3.5.2.3)
FU'(RZU f,(Q'x/s))~ R2U"f,(~l,/s)/Fi~o +1+'".
Thus (2.3.5.2.1) and (2.3.5.2.3) are isomorphisms or not together. But
(2.3.5.2.2)
and
(2.3.5.2.3)
have the same kernel, (F uin.~'u+li"~con ,"
( RzN
n
f, (Q*x/s)).
Since the source and target of (2.3.5.2.2) have the same
dimension over K, and the same is true of (2.3.5.2.2), it follows that
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 Fall '11
 NormanKatz
 Vector Space, Category theory, Isomorphism, Algebraic geometry

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