Quadratic and hermitian forms in additive and abelian categories

Abstract

During the last few years several papers concerned with the foundations of the theory of quadratic forms over arbitrary rings with involution have appeared. It is not necessary to give detailed references, in particular one thinks of the well known work of Bak [l], Bass [3], Karoubi, Knebusch [ll, 121, Ranicki, Vaserstein, and C. T. C. Wall. During the same period a number of problems quite similar to those occuring in the theory of quadratic forms were discussed, which, however, did not fit in the formalism developed so far. For example, one thinks of problems like the classification of pairs of forms, of sesquilinear forms, isometries , quadratic spaces with systems of subspaces, and also of quadratic forms over schemes, see e.g. [12, 13, 20, 22, 231. This situation called for a more general foundation of the theory of quadratic and hermitian forms. In this paper we try to give this foundation. Our basic object is an additive category &! together wit a duality functor *: & + A. In this situation one can define the most important notions of the theory of quadratic forms. Under suitable finiteness conditions one can prove a Krull-Schmidt theorem which is a sharpening of the classical Witt theorem. This result is basic for applications to the problems mentioned above. A preliminary version of this material is contained in [ 171. As just one application we discuss the classification of quadratic spaces with four subspaces. We hope that this disucssion will show clearly how one can solve a number of important classification problems of linear algebra. More applications can be found in [15, 16, 17, 22, 241. In the second part of the paper we discuss hermitian (not quadratic) forms in an abelian category. In an abelian category one has more structure, in particular one can introduce the notion of orthogonality. This allows one to introduce Grothendieck and Witt groups analogous to the G-groups in linear algebraic Ktheory which are obtained by factoring out exact sequences. As a basic result a Jordan-Holder theorem is proved for categories where all objects are of finite length. Using this theorem the computation of the Grothendieck group is 264