0, so by the exactness of the third column of Diagram 1.2 we see that dL(τ ) =
iL/F (γ) for some γ ⊕p∈XνF
(p)
(m 1). This means that ψ1([γ]) = ] so ψ1 is
surjective.
We now let µb : νF (m) H2
m+1
(F ) be defined by µb(ω) = bω. The exact
sequence of Aravire-Baeza shows that NL/F (νL(m)) = ker(µb). Let ˜b µ : νF (m)
H2
m+1(F
(X)) be the composite
˜b µ := iF
(X)/F
µb : νF (m) H2
m+1
(F ) H2
m+1
(F (X)).
Then NL/F (νL(m)) = ker(µb) ker(˜b) µ so we can define
νF (m)0 := ker(˜b)µ
and
νF (m)0 :=
νF (m)0
NL/F (νL(m))
.
Putting all this together gives the following.
Lemma 3.3. The map µb : νF (m)
Hm+1F
induces an isomorphism
µb : νF (m)0

=
ker
(
Hm+1F

Hm+1F
(X)
)
.
Proof. First we note by definition that µb : νF (m)
Hm+1F
maps νF (m)0
into
ker(Hm+1F

Hm+1F
(X)). As ker(µb) = NL/F (νL(m)) the induced map
νF (m)0 ker
(
Hm+1F

Hm+1F
(X)
)
is injective.
For surjectivity we suppose that α ker
(
Hm+1F

Hm+1F
(X)
)
. Then
iL/F (α) ker
(
Hm+1L

Hm+1L(X)
)
. However, L(X)/L is rational, so this latter
kernel is zero. It follows that iL/F (α) = 0. Hence by the exact sequence of Aravire
and Baeza we can express α = µb(ω) for some ω νF (m). By construction, such
ω νF (m)0. The surjectivity follows.
Lemma 3.4. Chasing in Diagram 1.2 gives a well-defined injective homomor-
phism
ψ2 : νF (m)0 νL(X)(m)0
defined by ψ2([ω]) = ] where NL(X)/F
(X)
) = iF
(X)/F
(ω).
Proof. Suppose that ω νF (m)0. Then by the exactness of the second column of
Diagram 1.2 there exists τ νL(X)(m) for which NL(X)/F (X)(τ ) = iF
(X)/F
(ω). We
claim that the class of τ in νL(X)(m)0 is uniquely determined by the class of ω in
νF (m)0. For suppose that ω νF (m)0, ] = [ω] νF (m)0, and τ νL(X)(m)
satisfies NL(X)/F
(X)
) = iF
(X)/F
). Then by the definition of νF (m)0, ω
ω = NL/F (ω0) for ω0 νL(m) and consequently NL(X)/F
(X)
(iL(X)/L(ω0) + τ ) =
iF
(X)/F
ω ) + iF
(X)/F
) = iF
(X)/F
(ω) = NL(X)/F
(X)
). So we find that
NL(X)/F
(X)
(iL(X)/L(ω0) + τ τ ) = 0 and by the exactness of the second column
of Diagram 1.2 we find iL(X)/L(ω0) + τ τ = iF
(X)/L(X)
(τ0) for τ0 νF
(X)
(m). It
follows that τ τ = iF
(X)/L(X)
(τ0) iL(X)/L(ω0), that is ] = ] νL(X)(m)0,
showing that ψ2 is well-defined.
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