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Derived Witt groups of a scheme. (English) Zbl 0972.18006
The paper defines Witt groups and $$L$$-groups of a triangulated category with a duality functor, e.g., a derived category of an exact category with duality, with the main application to the derived category of locally free sheaves of $$\mathcal{O} _X$$-modules of finite rank (i.e., (sheaves of sections of) vector bundles) over a Noetherian scheme $$X$$. This notion generalizes known concepts, as bilinear complexes, or Witt groups of complexes.
In particular the canonical functor from the category $$\mathcal{L} (X)$$ of vector bundles on $$X$$ to its derived category $$K(X)$$, which associates to a bundle $$\mathcal(E)$$ the (class of a) complex with unique nonzero term $$\mathcal(E)$$ in the degree $$0$$, induces a homomorphism $$i_X$$ from the (usual) Witt group of $$X$$ to the derived Witt group, i.e., the Witt group of the derived category $$K(X)$$.
The main result for an affine $$X = \text{Spec}(A)$$, where $$A$$ is a noetherian ring with $$2$$ invertible (theorem 4.29) states that the map $$i_A$$ is an isomorphism of the usual Witt group with the derived one.
The main reason to involve the triangulated category machinery is that there is a good notion of localization (factorization). The author constructs a three-term exact sequence of Witt groups of the given triangulated category with duality $$K$$, its localization $$S^{-1} K$$ and the full subcategory $$J(S)$$ of objects of $$K$$ isomorphic to $$0$$ in $$S^{-1} K$$ (theorem 5.17) and applies this result to a regular scheme and its open subscheme.

##### MSC:
 18E30 Derived categories, triangulated categories (MSC2010) 19G12 Witt groups of rings 11E81 Algebraic theory of quadratic forms; Witt groups and rings
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