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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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