τ0 νF
(X)
(m) with NL(X)/F
(X)
(iL(X)/L(ω0) + iL(X)/F
(X)
(τ0)) = iF
(X)/F
(ω). As
NL(X)/F
(X)
(iL(X)/F
(X)
(τ0)) = 0 we find
iF
(X)/F
(ω) = NL(X)/F
(X)
(iL(X)/L(ω0)) = iF
(X)/F
(NL/F (ω0)).
However, iF
(X)/F
: νF (m) νF
(X)
(m) is injective as F is algebraically closed in
F (X) so ω = NL/F (ω0). The injectivity of ψ2 follows.
The next two results require the computations outlined in Theorem 2.1.
Lemma 3.5. The map ψ2 : νF (m)0 νL(X)(m)0 in Lemma 3.4 is an isomor-
phism. Moreover, νF (m)0 = NL/F (νL(m)) + νF (m 1)
da
a
.
Proof. By Lemma 3.4 only surjectivity of ψ2 is required. Since ψ1 is surjec-
tive, by Theorem 2.1 parts (i) and (ii) we see that νL(X)(m)0 is generated by
the classes ] where τ =
∑s
i=1
αi KmL(X), as defined there. Further, by The-
orem 2.1 part (iii) we see that NL(X)/F
(X)
) = iF
(X)/F
) + for θ Tm
KmF and σ KmF (X). Now, ˜b([θ µ ]) = µb([iF
(X)/F
)]) = µb([iF
(X)/F
) +
]) = µb(NL(X)/F
(X)
)) = 0 H2
m+1(F
(X)), giving ] νF (m)0 and ] =
ψ2([θ ]). The surjectivity of ψ2 follows. But further, as θ Tm we see that
] NL/F (νL(m)) + νF (m 1)
da
a
. Since ψ2
−1
is an isomorphism the classes of
such θ generate νF (m)0, and the second assertion follows.
Putting the previous five lemmas together gives the next result.
Theorem 3.6. We have two isomorphisms
ψ2
−1
ψ1 : H0 1(X, ν(m))

=
νF (m)0,
˜b µ : νF (m)0

=
ker
(
Hm+1F

Hm+1F
(X)
)
Moreover, ker
(
Hm+1F

Hm+1F
(X)
)
= νF (m 1)
(
b
da
a
)
H2
m+1
F.
Proof. The first isomorphism is the composition of the isomorphisms given in
Lemmas 3.2 and 3.5. The second is given in Lemma 3.3. For the final statement,
by Lemma 3.5 we have νF (m)0 = NL/F (νL(m)) + νF (m 1)
da
a
so in particular
νF (m)0 is generated by classes in νF (m 1)
da
a
. The result follows applying ˜b. µ
Theorem 1.5 now follows composing the isomorphisms specified in Theorem
3.6. Theorem 1.6 of Aravire and Baeza can be obtained as follows. Suppose that)
θ
ker(ImWqF

ImWqF
(X)) and suppose λ ker
(
Hm+1F −→Hn+1F (X)
corresponds to θ under Kato’s isomorphism
ImWqF
=
Hm+1F
. Applying Theorem
3.6 we have λ = (dlog θ0
(
b
da
a
)
) for θ0 Km−1F . So we conclude by Kato’s
isomorphism yet again that λ corresponds to an element ψ0 a, b]]
ImWqF
.
Theorem 1.6 follows.
Aside from the discussions and calculations that give Lemma 1.3, Lemma 2.2
and Lemma 2.4, the paper is complete. These three results are proved in the next
section.
Previous Page Next Page