H1(X, ν) of Conics and Witt Kernels in Characteristic 2

Roberto Aravire and Bill Jacob

Abstract. Suppose that F is a ﬁeld of characteristic two, φ = [1, b] ⊥ a

is a Pﬁster neighbor of a, b]], and X is the conic deﬁned by φ = 0. The

groups H0

1(X,

ν(m)) are computed for m ≥ 1. This provides a geometric

interpretation of the recent result of Aravire and Baeza that I

m

Wq (F (φ)/F ) =

I

m−1

F · a, b]].

Suppose F is a ﬁeld. The Witt kernels W (F (φ)/F ) where φ is a quadratic form

and F (φ) is its generic zero ﬁeld have an important history in the algebraic theory

of quadratic forms. Their behavior is closely linked to the K-cohomology of the

quadric hypersurface X deﬁned by φ = 0. Of particular interest is the case where

φ in an anisotropic Pﬁster neighbor, largely because of its role in the proof of the

Milnor conjecture. In this paper the case where the characteristic of F is two and

φ := [1, b] ⊥ a is a Pﬁster neighbor of a, b]] is considered in detail. The main goal

is computation of the groups H0 1(X, ν(m)). This provides a geometric interpretation

of the recent result of Aravire and Baeza that I

m

Wq(F (X)/F ) = I

m−1

F · a, b]],

which is readily obtained as a corollary.

In the ﬁrst section we give the basic deﬁnitions and state the main results. In

the second section we lay out the computational lemmas that are needed to prove

the main results. Although the main theorem involves Milnor K-theory mod two,

one of the key points of the paper is that it is necessary to compute with the full

Milnor K-groups. For the results depend heavily upon Izboldin’s Theorem [I, Th. A

p 129] that in characteristic p the Milnor K-theory of a ﬁeld has no p-torsion, as well

as certain representations of elements in 2KnF which would vanish if computing

only mod two.

In the third section the proofs of the main results are given, using the results

from the ﬁrst and second sections. For the most part, this section is devoted to

checking that the maps derived from the calculations in section two are well-deﬁned.

The fourth section collects the technical calculations from the ﬁrst two sections that

were deferred in order to help the reader see the overall flow of the argument.

1991 Mathematics Subject Classiﬁcation. Primary 11E81, 11E70; Secondary 11G99.

Key words and phrases. K-theory, quadratic forms.

This work has been supported by Fondecyt 1050 337, Univ. A. Prat. (ﬁrst author) and

Proyecto Anillos, PBCT, ACT05 (ﬁrst and second authors).

1

1

Roberto Aravire and Bill Jacob

Abstract. Suppose that F is a ﬁeld of characteristic two, φ = [1, b] ⊥ a

is a Pﬁster neighbor of a, b]], and X is the conic deﬁned by φ = 0. The

groups H0

1(X,

ν(m)) are computed for m ≥ 1. This provides a geometric

interpretation of the recent result of Aravire and Baeza that I

m

Wq (F (φ)/F ) =

I

m−1

F · a, b]].

Suppose F is a ﬁeld. The Witt kernels W (F (φ)/F ) where φ is a quadratic form

and F (φ) is its generic zero ﬁeld have an important history in the algebraic theory

of quadratic forms. Their behavior is closely linked to the K-cohomology of the

quadric hypersurface X deﬁned by φ = 0. Of particular interest is the case where

φ in an anisotropic Pﬁster neighbor, largely because of its role in the proof of the

Milnor conjecture. In this paper the case where the characteristic of F is two and

φ := [1, b] ⊥ a is a Pﬁster neighbor of a, b]] is considered in detail. The main goal

is computation of the groups H0 1(X, ν(m)). This provides a geometric interpretation

of the recent result of Aravire and Baeza that I

m

Wq(F (X)/F ) = I

m−1

F · a, b]],

which is readily obtained as a corollary.

In the ﬁrst section we give the basic deﬁnitions and state the main results. In

the second section we lay out the computational lemmas that are needed to prove

the main results. Although the main theorem involves Milnor K-theory mod two,

one of the key points of the paper is that it is necessary to compute with the full

Milnor K-groups. For the results depend heavily upon Izboldin’s Theorem [I, Th. A

p 129] that in characteristic p the Milnor K-theory of a ﬁeld has no p-torsion, as well

as certain representations of elements in 2KnF which would vanish if computing

only mod two.

In the third section the proofs of the main results are given, using the results

from the ﬁrst and second sections. For the most part, this section is devoted to

checking that the maps derived from the calculations in section two are well-deﬁned.

The fourth section collects the technical calculations from the ﬁrst two sections that

were deferred in order to help the reader see the overall flow of the argument.

1991 Mathematics Subject Classiﬁcation. Primary 11E81, 11E70; Secondary 11G99.

Key words and phrases. K-theory, quadratic forms.

This work has been supported by Fondecyt 1050 337, Univ. A. Prat. (ﬁrst author) and

Proyecto Anillos, PBCT, ACT05 (ﬁrst and second authors).

1

1