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]) = [τ ] 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]) = [τ ] where NL(X)/F
(X)
(τ ) = 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 ] = [ω] ∈ νF (m)0, and τ ∈ νL(X)(m)
satisfies NL(X)/F
(X)
(τ ) = iF
(X)/F
(ω ). 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 − ω ) + iF
(X)/F
(ω ) = iF
(X)/F
(ω) = NL(X)/F
(X)
(τ ). 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 [τ ] = [τ ] ∈ νL(X)(m)0,
showing that ψ2 is well-defined.
iL/F (γ) for some γ ∈ ⊕p∈XνF
(p)
(m − 1). This means that ψ1([γ]) = [τ ] 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]) = [τ ] where NL(X)/F
(X)
(τ ) = 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 ] = [ω] ∈ νF (m)0, and τ ∈ νL(X)(m)
satisfies NL(X)/F
(X)
(τ ) = iF
(X)/F
(ω ). 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 − ω ) + iF
(X)/F
(ω ) = iF
(X)/F
(ω) = NL(X)/F
(X)
(τ ). 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 [τ ] = [τ ] ∈ νL(X)(m)0,
showing that ψ2 is well-defined.
