## Quadratic forms, orderings and quaternion algebras over rings with many units

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##### Date

1988##### Author

Walter, Leslie J.

##### Type

Thesis##### Degree Level

Masters##### Abstract

The "algebraic" theory of quadratic forms over fields of characteristic â‰ 2 dates back to the 1937
paper of Witt [37]. It was in this paper that the Witt ring of a field was first considered. Thirty
years later, Pfister published his work on the structure of the Witt ring [31]-[33]. These papers
triggered an enormous growth in the theory of quadratic forms. Both [15] and [35] are excellent
references for the field case.
By the early 1970's, much of the algebraic theory of quadratic forms was extended to semilocal
rings. (See [3] or [23]. In [3] the characteristic 2 case is also considered.) Since then the field has
branched out in many directions. Of particular interest here are the abstract theories of Knebusch,
Rosenberg and Ware [13], [14], and of Marshall and Yucas [24], [22]. The study of the Witt ring
modulo its nilradical has led to the development of a reduced theory of quadratic forms [4], [5], [11].
The theory of abstract spaces of orderings contained in Marshall's series of papers [17]-[21] provides
an axiomatic treatment of this reduced theory.
In this thesis, we show all of these abstract theories can be applied to the study of quadratic forms
over a ring with many units with 2 âˆˆ A*. This work was started by McDonald and Kirkwood in
[26]. However, they considered only those rings with many units which had infinite residue fields.
As the title suggests, we are also interested in orderings and quaternion algebras. We only consider
these subjects in so far as they relate to the study of quadratic forms.
The first two chapters are introductory in nature. The results of the first chapter are taken from
a paper by McDonald and Waterhouse [27]. For more on modules over rings with many units, we
refer the reader to [8], (9]. In chapter 2, we introduce the notions of quadratic forms and the Witt
ring of an arbitrary commutative ring. A good reference for this material is [28].
The most important result of the first two chapters is (1.2.2), that is, over a ring with many units,
every finitely generated projective module of constant rank is free. Consequently, we can (and do)
restrict our attention to quadratic forms based on free modules in what follows.
The third chapter covers the "standard" theory of quadratic forms. The main results are (3.1.4)
and (3.2.4). Once these are available, the results of sections 3, 4, and 5 are obtained just as in the
case of semilocal rings. Of special importance is (3.3.1). This shows W (A) is an abstract Witt ring
in the sense of Knebusch, Rosenberg and Ware. We apply this in chapter 5 in the study of the
structure of W (A).
The purpose of chapter 4 is to provide a brief introduction to the concept of orderings on rings
with many units. Only those results needed in later chapters are given. For a broader treatment of
orderings on commutative rings, we refer the reader to [6], [16], [23].
The first two sections of chapter 5 are a direct consequence of the abstract theory in [13], [14], [12]
and we freely make use of the results contained therein. Sections 3 and 4 develop the connection
between the space of signatures of the Witt ring and the real spectrum of the base ring. In section 5,
we show that every preordering T âŠ† A gives rise to a space of orderings (XT, A* /T*) in the sense of
[17]-[21]. Generalizing a result in [11], in section 6 we show every non-trivial fan on A arises locally
from a fan on some residue field of A. As a result of this and the representation theorem in [20], we
show the problem of describing the Witt ring WT(A) as a subring of Cont(XT, Z) may be reduced
to the field case. In section 7, a local-global criterion for T-isotropy is obtained. This generalizes
another result in [11] and is just a special case of a more general version given in [25].
Chapter 6 deals with the Brauer group and quaternion algebras. The main result of this chapter
is the "cancellation" theorem (6.1.15). This was shown to hold for connected rings with many units
in [10]. The general version given here is due to Marshall. Section 2 relates quaternion algebras to
quadratic forms and shows that the abstract theory in [24] and [22] may be applied here. In section
3, we use the results of the first two sections to obtain the Hasse invariant of a quadratic form. This
is done just as in the abstract case in [22] except we may use Br2(A) as our target group.
The notation used throughout is for the most part standard. All rings considered are commutative
with 1. For a ring A and an ideal a âŠ† A, xÌ„ is used to denote the coset x + a âŠ† A/a if no confusion
will result. If p âŠ† A is a prime ideal, we denole the localization of A at p by Ap. A* denotes the
group of units of a ring A, J(A) denotes the Jacobson radical of A and Nil(A) denotes the nilradical
of A. We assume the reader is familiar with basic commutative algebra as given in, for example, [1].