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Abstract Witt rings. (English) Zbl 0451.10013
Queen’s Pap. Pure Appl. Math. 57, 257 p. (1980).
The book presents an account of the abstract theory of quadratic forms. The first chapter covers the fundamentals of the theory of quadratic forms over a field and motivates the author’s axiomatization of the theory by means of quaternionic structures. In the second and third chapters the author introduces the abstract quaternionic structure \((G,Q,q)\) and shows how to develop an abstract theory of quadratic forms associated to it, so that forms, isometry of forms, Witt rings, Pfister forms, Hasse and Witt invariants are defined and studied. This approach comes from an earlier joint paper by the author and J. Yucas [Linked quaternionic mappings and their associated Witt rings, Pac. J. Math. 95, 411–425 (1981; Zbl 0423.10013)].
In the fourth chapter the author shows how to introduce the abstract theory in another way using a new system of axioms in terms of Witt rings. It is shown that every abstract Witt ring \(R\) determines the quaternionic structure \((G_R,Q_R,q_R)\) and the theories defined by quaternionic structures and by abstract Witt rings are equivalent.
The next chapter contains many fine results on finitely generated Witt rings. It is proved here that every such Witt ring can be expressed in terms of \(\mathbb Z/2\mathbb Z\) and some so-called basic indecomposables using the operations of group ring formation and direct product. Moreover, Witt rings \(R\) with \(\vert G_R\vert < \infty\) and \(\vert Q_R\vert < 4\) are fully classified.
The chapters VI–IX are devoted to reduced Witt rings, that is, Witt rings with trivial nilradical. There is a direct relation between reduced Witt rings and spaces of orderings known from the series of author’s papers [Can. J. Math. 31, 320–330 (1979; Zbl 0412.10012); Can. J. Math. 31, 604–616 (1979; Zbl 0419.10024); ibid. 32, 603–627 (1980; Zbl 0433.10009); Trans. Am. Math. Soc. 258, 505–521 (1980; Zbl 0427.10015)].
The author presents his own most important results on spaces of orderings expressed here in terms of reduced Witt rings. For example, in chapter VI one can find a nice structure theorem for finitely generated reduced Witt rings known from the first of the above quoted author’s papers. In chapter VIII the author constructs sheaves of reduced Witt rings and he proves that a reduced Witt ring whose space of orderings has only finitely many accumulation points is built up from finitely generated reduced Witt rings in a finite number of steps by using the operations of group ring formation and sheaf formation.
In the last chapter the author shows that the theory of quadratic forms over a semilocal ring \(A\) determines uniquely a quaternionic structure with the associated form theory coinciding with the theory of quadratic forms over \(A\).
The book seems to be exceptionally carefully written. It includes the most important results of the subject not available in print elsewhere. The book will be very useful to all those readers – beginners as well as experts – who are interested in abstract theory of quadratic forms.

MSC:
11E81 Algebraic theory of quadratic forms; Witt groups and rings
11E04 Quadratic forms over general fields
11E08 Quadratic forms over local rings and fields
12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.)
11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
11-02 Research exposition (monographs, survey articles) pertaining to number theory