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



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The "algebraic" theory of quadratic forms over fields of characteristic ≠ 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 ∈ 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 ⊆ 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 ⊆ A, x̄ is used to denote the coset x + a ⊆ A/a if no confusion will result. If p ⊆ 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].





Master of Science (M.Sc.)







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