We denote by ℘(x) = x2 +x the characteristic two Artin-Schreier operator. Let

X be the conic deﬁned by the Pﬁster neighbor φ := [1, b] ⊥ a of an anisotropic

quadratic Pﬁster form a, b]]. So the curve X has an aﬃne equation y2 + y + b +

ax2 = 0 and the function ﬁeld of X is F (X) = F (x, y) where ℘(y) = b + ax2. Let

L be the separable quadratic extension L = F (β) where ℘(β) = b ∈ F . Then φ

becomes isotropic over L and the curve XL is rational.

All K-theory groups will be Milnor K-theory. The groups νF (m) and H2

m+1(F

)

are deﬁned by the exact sequence (see [BK])

0 → νF (m) → ΩF

m

℘

→ ΩF

m

/dF → H2

m+1

(F ) → 0.

℘ : ΩF

m

→ ΩF

m/dF

is deﬁned by ℘(a

df1

f1

∧ ··· ∧

dfm

fm

) =

(ap

− a)

df1

f1

∧ ··· ∧

dfm

fm

and

νF (m) is ker(℘). Since the characteristic of F is two the Theorems of Kato [K]

asserts that dlog : KmF/2KmF

∼

=

→ νF (m) is an isomorphism and that H2

m+1(F

)

∼

=

ImWqF/Im+1WqF . The fact that KmF/2KmF

∼

=

νF (m) → ΩF m has one im-

mediate and particularly important consequence; if L is a separable extension of

F we necessarily have νF (m) → νL(m) is injective as a 2-basis for F remains

2-independent in L.

We use the localization sequences which deﬁne the (Milnor) K-cohomology

groups,

0 →

H0(X,

Km) → KmF (X) →

p∈X

Km−1F (p) →

H1(X,

Km) → 0

where the map dF : KmF (X) →

p∈X

Km−1F (p) is the sum of tame symbols

corresponding to the valuations vp : F (p) → Z at each point p ∈ X. Since XL is

rational, results of Milnor [Mi] show that

Hi(XL,

Km)

∼

=

Km−iL for i = 0, 1. When

i = 0 the isomorphism is induced by L → L(X) and the second is given by the sum

of norms ΣN : ⊕q∈XL Km−1L(q) → Km−1L.

The localization sequences for X and XL ﬁt into the commutative Diagram 1.1

below.

KmF (X) →

p∈X

Km−1F (p)

↓ ↓

KmL → KmL(X) →

q∈XL

Km−1L(q) → Km−1L

↓ ↓ ↓

KmF → KmF (X) →

p∈X

Km−1F (q)

Diagram 1.1.

We remark that in the third column of Diagram 1.1, whenever there is a single

q ∈ XL above a point p ∈ X, then L(q) is a proper quadratic extension of F (p) and

the ﬁrst map in the column is induced by inclusion while the second is the norm.

In case there are two q ∈ XL above a point p ∈ X, then L(q) = F (p) and the maps

are the diagonal and codiagonal. The calculations given in Section 2 take place

within this diagram. The middle row of Diagram 1.1 is exact by Milnor’s result,

but the columns and other row need not be. However, since L/F is quadratic, the

X be the conic deﬁned by the Pﬁster neighbor φ := [1, b] ⊥ a of an anisotropic

quadratic Pﬁster form a, b]]. So the curve X has an aﬃne equation y2 + y + b +

ax2 = 0 and the function ﬁeld of X is F (X) = F (x, y) where ℘(y) = b + ax2. Let

L be the separable quadratic extension L = F (β) where ℘(β) = b ∈ F . Then φ

becomes isotropic over L and the curve XL is rational.

All K-theory groups will be Milnor K-theory. The groups νF (m) and H2

m+1(F

)

are deﬁned by the exact sequence (see [BK])

0 → νF (m) → ΩF

m

℘

→ ΩF

m

/dF → H2

m+1

(F ) → 0.

℘ : ΩF

m

→ ΩF

m/dF

is deﬁned by ℘(a

df1

f1

∧ ··· ∧

dfm

fm

) =

(ap

− a)

df1

f1

∧ ··· ∧

dfm

fm

and

νF (m) is ker(℘). Since the characteristic of F is two the Theorems of Kato [K]

asserts that dlog : KmF/2KmF

∼

=

→ νF (m) is an isomorphism and that H2

m+1(F

)

∼

=

ImWqF/Im+1WqF . The fact that KmF/2KmF

∼

=

νF (m) → ΩF m has one im-

mediate and particularly important consequence; if L is a separable extension of

F we necessarily have νF (m) → νL(m) is injective as a 2-basis for F remains

2-independent in L.

We use the localization sequences which deﬁne the (Milnor) K-cohomology

groups,

0 →

H0(X,

Km) → KmF (X) →

p∈X

Km−1F (p) →

H1(X,

Km) → 0

where the map dF : KmF (X) →

p∈X

Km−1F (p) is the sum of tame symbols

corresponding to the valuations vp : F (p) → Z at each point p ∈ X. Since XL is

rational, results of Milnor [Mi] show that

Hi(XL,

Km)

∼

=

Km−iL for i = 0, 1. When

i = 0 the isomorphism is induced by L → L(X) and the second is given by the sum

of norms ΣN : ⊕q∈XL Km−1L(q) → Km−1L.

The localization sequences for X and XL ﬁt into the commutative Diagram 1.1

below.

KmF (X) →

p∈X

Km−1F (p)

↓ ↓

KmL → KmL(X) →

q∈XL

Km−1L(q) → Km−1L

↓ ↓ ↓

KmF → KmF (X) →

p∈X

Km−1F (q)

Diagram 1.1.

We remark that in the third column of Diagram 1.1, whenever there is a single

q ∈ XL above a point p ∈ X, then L(q) is a proper quadratic extension of F (p) and

the ﬁrst map in the column is induced by inclusion while the second is the norm.

In case there are two q ∈ XL above a point p ∈ X, then L(q) = F (p) and the maps

are the diagonal and codiagonal. The calculations given in Section 2 take place

within this diagram. The middle row of Diagram 1.1 is exact by Milnor’s result,

but the columns and other row need not be. However, since L/F is quadratic, the