Grundlehren der Mathematischen Wissenschaften. 294. Berlin etc.: Springer-Verlag. xi, 524 p. (1991).
Until thirty years ago the theory of quadratic forms was normally understood to deal with forms over rings of integers (and their fields of fractions). Then the algebraic theory developed, centering around the computation of the Witt group of quadratic forms over general fields. The development was influenced by that of algebraic $K$-theory, and like $K$- groups one soon began to study also Witt groups for general rings. Although this hermitian $K$-theory is not the proper subject here, the present book is related to it rather than to the algebraic or integral theory. As one knows, Serre’s conjecture (now the Quillen-Suslin theorem) on projective modules over polynomial rings has played a major role in the development of $K$-theory. The corresponding problem on quadratic spaces over polynomial rings has led to much of what one finds in the second half of this book; a considerable portion of these results is due to Knus, Ojanguren, Parimala and Sridharan.
The first half of the book (Chapters I to IV) treats the foundations of quadratic and hermitian forms in a very general setting. In particular, assuming 2 to be invertible in the ground ring is normally avoided. The way to the later applications is indicated already by the final section of Chapter I on “patching” of forms. Chapter II presents the categorical view of quadratic forms, adding some applications to what can be found also in Scharlau’s book on the algebraic theory. Otherwise the contents of both books are rather disjoint, except perhaps for some of the material on Clifford algebras and invariants. As Knus works over general commutative rings, his treatment of this material, however, is much more delicate. It is based, of course, on the notion of Azumaya algebra introduced in Chapter III. This large chapter, exposing the technique of faithfully flat descent and corresponding cohomological tools, is essential especially for the classification of forms of low rank (up to 6) in Chapter V, one of the main goals of the book. Applications are also made later in the context of polynomial rings.
Chapter VI introduces $K$-theory, giving splitting, stability and cancellation theorems both for projective modules and forms. As in the whole book, careful and complete proofs can be found, including a new one for unitary stability in Section 4. Results from this section have to be used at a crucial point in the next chapter on polynomial rings, namely in the proof of the local Horrocks theorem. This theorem, when combined with the patching method, gives both the Quillen-Suslin theorem and its quadratic analogue (due to Ojanguren and Suslin-Kopeiko) which says that for a field $k$ of characteristic not 2 any isotropic quadratic space over $\bbfA\sp n(k)$ is extended from $k$. The chapter also includes some older results on the subject, a more general version of Ojanguren’s theorem, the results on non-extended anisotropic spaces, and something on quadratic spaces over $\bbfP\sp n(k)$ where the author deviates from his general principle to deal only with affine schemes.
The final Chapter VIII on Witt groups of affine rings returns to this principle, although some general notions and cited results are concerned also with other schemes. The main problem treated here is the injectivity of the map from the Witt group of a domain to that of its quotient field. Furthermore, one finds some instructive computations for specific curves and surfaces. The study of forms over algebraic varieties seems to have only begun, and the present book will be of great value for its promotion.