Witt Groups

In his 1937 paper [86], Ernst Witt introduced a group structure – and even a ring structure – on the set of isometry classes of anisotropic quadratic forms, over an arbitrary field k. This object is now called the Witt group W(k) of k. Since then, Witt’s construction has been generalized from fields to rings with involution, to schemes, and to various types of categories with duality. For the sake of efficacy, we review these constructions in a non-chronological order. Indeed, in Sect. 1.2, we start with the now “classical framework” in its most general form, namely over exact categories with duality. This folklore material is a basically straightforward generalization of Knebusch’s scheme case [41], where the exact category was the one of vector bundles. Nevertheless, this level of generality is hard to find in the literature, like e.g. the “classical sublagrangian reduction” of Sect. 1.2.5. In Sect. 1.3, we specialize this classical material to the even more classical examples listed above: schemes, rings, fields. We include some motivations for the use of Witt groups.