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).
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