port among the points with π-degree 1. Alternately, any class can be represented by

a cycle with support among the points Qr, where r ∈ F or Q∞. By a change of vari-

ables, we can move away from Q∞. The same applies to classes in H1(X, ν(m)).

Lemma 1.3 is proved in Section 4. A reﬁnement of this lemma for elements of

H0

1(X,

ν(m)) is given next.

Lemma 1.4. After a change of variables, any class [γ] ∈ H0

1(X,

ν(m)) can be

represented by

γ =

s

i=1

(Qri ; {mi(

i

+

√

pri )} ⊗ χi) ∈ ⊕p∈XKm−1F (p)

for ri, mi,

i

∈ F and χi ∈ Km−2F . Moreover,

∑s

i=1

(

{mi

2( 2

i

+ pri )} ⊗ χi

)

=

2χ∞ ∈ Km−1F for some χ∞ ∈ Km−1F .

Proof. As F (Qri ) = F (

√

pri ) is quadratic, Km−1F (Qri ) = K1F (

√

pri ) ⊗ Km−2F .

Therefore by Lemma 1.3, it is possible to represent [γ] by γ ∈ ⊕p∈XKm−1F (p)

where

γ =

s

i=1

(Qri ; {mi(

i

+

√

pri )} ⊗ χi)

with ri, mi,

i

∈ F and χi ∈ Km−2F . Since iL/F ([γ]) is zero in

H1(XL,

ν(m)) we

can express iL/F (γ) = dLτ +2γ for some τ ∈ KmL(X) and γ ∈

q∈XL

Km−1L(q).

Using the fact that ΣN(dL(τ )) = 0 ∈ Km−1L we ﬁnd 2ΣN(γ ) = ΣN(iL/F (γ)) =

∑s

i=1

(

{mi

2( 2

i

+ pri )} ⊗ χi

)

∈ 2KmL. Moreover, as L is separable over F we know

that

Km−1F/2Km−1F

∼

= νF (m − 1)

iL/F

→ νL(m − 1)

∼

= Km−1L/2Km−1L

is injective. So as

∑s

i=1

(

{mi 2( 2

i

+ pri )} ⊗ χi

)

∈ Km−1F we must in fact have

∑s

i=1

(

{mi 2( 2

i

+ pri )} ⊗ χi

)

∈ 2Km−1F . This gives the lemma.

The main result of the paper is the following.

Theorem 1.5. If φ = [1, b] ⊥ a is a Pﬁster neighbor of a, b]], and if X is

the conic which is given by φ = 0, then there exists an isomorphism

ψ : H0

1(X,

ν(m))

∼

=

→ νF (m − 1) ∧ b

da

a

⊆ H2

m+1F.

As an application we will be able to work with Diagram 1.2 and obtain the

following result of Aravire and Baeza.

Theorem 1.6. (Aravire and Baeza) The kernel on the graded Witt groups,

ker(ImWqF

→

ImWqF

( a, b]])) =

Im−1F

· a, b]].

When the characteristic of F is diﬀerent from 2, the analogue of Theorem 1.6

when m = 3 is due to Arason [A]. In the case of a 3-fold Pﬁster form, and when

m = 4, Jacob and Rost [JR] obtained the analogue of this result using an analysis

of the K-cohomology of a 3-dimensional quadric hypersurface. Computations of

the K-cohomology of quadrics are also crucial to the work of Voevodsky on the

a cycle with support among the points Qr, where r ∈ F or Q∞. By a change of vari-

ables, we can move away from Q∞. The same applies to classes in H1(X, ν(m)).

Lemma 1.3 is proved in Section 4. A reﬁnement of this lemma for elements of

H0

1(X,

ν(m)) is given next.

Lemma 1.4. After a change of variables, any class [γ] ∈ H0

1(X,

ν(m)) can be

represented by

γ =

s

i=1

(Qri ; {mi(

i

+

√

pri )} ⊗ χi) ∈ ⊕p∈XKm−1F (p)

for ri, mi,

i

∈ F and χi ∈ Km−2F . Moreover,

∑s

i=1

(

{mi

2( 2

i

+ pri )} ⊗ χi

)

=

2χ∞ ∈ Km−1F for some χ∞ ∈ Km−1F .

Proof. As F (Qri ) = F (

√

pri ) is quadratic, Km−1F (Qri ) = K1F (

√

pri ) ⊗ Km−2F .

Therefore by Lemma 1.3, it is possible to represent [γ] by γ ∈ ⊕p∈XKm−1F (p)

where

γ =

s

i=1

(Qri ; {mi(

i

+

√

pri )} ⊗ χi)

with ri, mi,

i

∈ F and χi ∈ Km−2F . Since iL/F ([γ]) is zero in

H1(XL,

ν(m)) we

can express iL/F (γ) = dLτ +2γ for some τ ∈ KmL(X) and γ ∈

q∈XL

Km−1L(q).

Using the fact that ΣN(dL(τ )) = 0 ∈ Km−1L we ﬁnd 2ΣN(γ ) = ΣN(iL/F (γ)) =

∑s

i=1

(

{mi

2( 2

i

+ pri )} ⊗ χi

)

∈ 2KmL. Moreover, as L is separable over F we know

that

Km−1F/2Km−1F

∼

= νF (m − 1)

iL/F

→ νL(m − 1)

∼

= Km−1L/2Km−1L

is injective. So as

∑s

i=1

(

{mi 2( 2

i

+ pri )} ⊗ χi

)

∈ Km−1F we must in fact have

∑s

i=1

(

{mi 2( 2

i

+ pri )} ⊗ χi

)

∈ 2Km−1F . This gives the lemma.

The main result of the paper is the following.

Theorem 1.5. If φ = [1, b] ⊥ a is a Pﬁster neighbor of a, b]], and if X is

the conic which is given by φ = 0, then there exists an isomorphism

ψ : H0

1(X,

ν(m))

∼

=

→ νF (m − 1) ∧ b

da

a

⊆ H2

m+1F.

As an application we will be able to work with Diagram 1.2 and obtain the

following result of Aravire and Baeza.

Theorem 1.6. (Aravire and Baeza) The kernel on the graded Witt groups,

ker(ImWqF

→

ImWqF

( a, b]])) =

Im−1F

· a, b]].

When the characteristic of F is diﬀerent from 2, the analogue of Theorem 1.6

when m = 3 is due to Arason [A]. In the case of a 3-fold Pﬁster form, and when

m = 4, Jacob and Rost [JR] obtained the analogue of this result using an analysis

of the K-cohomology of a 3-dimensional quadric hypersurface. Computations of

the K-cohomology of quadrics are also crucial to the work of Voevodsky on the