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 [τ ] 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)

(τ ) = iF

(X)/F

(θ ) + 2σ for θ ∈ Tm ⊆

KmF and σ ∈ KmF (X). Now, ˜b([θ µ ]) = µb([iF

(X)/F

(θ )]) = µb([iF

(X)/F

(θ ) +

2σ ]) = µb(NL(X)/F

(X)

(τ )) = 0 ∈ H2

m+1(F

(X)), giving [θ ] ∈ νF (m)0 and [τ ] =

ψ2([θ ]). The surjectivity of ψ2 follows. But further, as θ ∈ Tm we see that

[θ ] ∈ 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.

(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 [τ ] 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)

(τ ) = iF

(X)/F

(θ ) + 2σ for θ ∈ Tm ⊆

KmF and σ ∈ KmF (X). Now, ˜b([θ µ ]) = µb([iF

(X)/F

(θ )]) = µb([iF

(X)/F

(θ ) +

2σ ]) = µb(NL(X)/F

(X)

(τ )) = 0 ∈ H2

m+1(F

(X)), giving [θ ] ∈ νF (m)0 and [τ ] =

ψ2([θ ]). The surjectivity of ψ2 follows. But further, as θ ∈ Tm we see that

[θ ] ∈ 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.