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
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