Derived Witt groups of a scheme

Abstract

The Witt group of a triangulated category with duality is the quotient of the monoid of symmetric spaces by the submonoid of neutral forms. Neutral forms are defined in a traditional way, using lagrangians. To any noetherian scheme X is associated a derived category with duality, denoted by K( X). The Witt group of K( X) will be called the (derived) Witt group of X. There is an isomorphism between the usual Witt group of a ring in which 2 is a unit and its derived Witt group. This approach allows us to compute the kernel of W( A)→W( Q), where A is a domain and Q its field of fractions. This kernel turns out to be the Witt group of some suitable triangulated category with duality. The point of view of derived categories seems particularly useful for localization. Let U be an open subscheme of a regular scheme X. It is not hard to establish that K( U) is a localization of K( X) with respect to a suitable multiplicative system. Denote by J the full subcategory of K( X) on the objects vanishing in K( U). We construct a connecting homomorphism from the Witt group of K( U) to some Witt group of J, associating skew-symmetric forms to symmetric ones. We prove that the kernel of this homomorphism is precisely the part of W(K( U)) coming from X. Using these results we obtain a very simple proof of purity in dimension 3.