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.