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The Mayer-Vietoris principle for Grothendieck-Witt groups of schemes. (English) Zbl 1193.19005
For a scheme $$X$$ let $$GW_{0}(X)$$ denote the Grothendieck-Witt group of symmetric bilinear spaces over $$X$$. This is the abelian group generated by isometry classes $$[\mathcal V,\phi ]$$ of vector bundles $$\mathcal V$$ over $$X$$ with a nonsingular symmetric bilinear form $$\phi: \mathcal V\otimes \mathcal V\to O_X$$ subject to the relations $$[(\mathcal V,\phi) \perp (\mathcal V ',\phi ' ]= [{\mathcal V},{\phi}]+[\mathcal V ',\phi ']$$ and $$[\mathcal M,\phi]=[\mathcal H(\mathcal N)]$$ for every metabolic space $$(\mathcal M, \phi)$$ with Lagrangian subbundle $$\mathcal N=\mathcal N^{\perp} \subset \mathcal M$$ and associated hyperbolic space $$\mathcal H(\mathcal N)$$. The higher Grothendieck-Witt groups $$GW_{i}(X), \, i\in {\mathbb N}$$ were defined by the author [“Higher Grothendieck-Witt groups of exact categories”, J. K-theory (to appear)].
In the paper the author proves the Mayer-Vietoris sequence for open covers (Theorem 1). This main theorem of the paper (in fact Theorem 16 of the paper is more general and includes versions for skew symmetric forms and coefficients in line bundles different than $$O_{X}$$ ) is derived from the localization (Theorem 2) and Zariski excision (Theorem 3) theorems. The author proves also additivity, fibration and approximation theorems for the hermitian $$K$$-theory of exact categories with weak equivalences and duality. As the author noticed, P. Balmer [K-Theory 23, No. 1, 15–30 (2001; Zbl 0987.19002)] and J. Hornbostel [Topology 44, No. 3, 661–687 (2005; Zbl 1078.19004)] proved similar results to theorems 1–3. However, their assumptions are stricter than these of the author.

MSC:
 19G38 Hermitian $$K$$-theory, relations with $$K$$-theory of rings 19E08 $$K$$-theory of schemes 19G12 Witt groups of rings 11E81 Algebraic theory of quadratic forms; Witt groups and rings 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry
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