Koszul complexes and symmetric forms over the punctured affine space. (English) Zbl 1078.11026

The paper contains computation of triangulated Witt groups of affine space with origin removed. It would be much easier to review papers on derived Witt groups if authors provided us with a book as a standard reference. In view of lack of standard reference let us recall following. The derived Witt groups, introduced by the first author P. Balmer [\(K\)-theory 19, 311–363 (2000; Zbl 0953.18003), Math. Z. 236, 351–382 (2001; Zbl 1004.18010)], form a \(4\)-periodic sequence \(W^{i}(X)\) of groups functorially attached to a \({\mathbb Z}\left[ {1\over 2}\right] \)-scheme \(X\); more precisely the groups \(W^{i}(X)\) depend on the derived category of the category of vector bundles on \(X\). Moreover, \(W^{0}(X)\) is (canonically isomorphic to) the usual Witt group of classes of (vector bundles with) symmetric bilinear forms, \(W^{2}(X)\) is (canonically isomorphic to) the usual Witt group of classes of (vector bundles with) skew-symmetric bilinear forms, \(W^{1}(X)\) and \(W^{3}(X)\) are \(L\)-groups of \( X \). There is a graded multiplicative structure on \(W^{\bullet }(X)\). If there is a closed subscheme \(Y\) in \(X\), then the second author studied [S. Gille, Math. Z. 244, 211–233 (2003; Zbl 1028.11025), Math. Ann. 322, 103–137 (2002; Zbl 1010.19003)] another \(4\)-periodic sequence of groups - Witt groups \(W_{Y}^{i}(X)\) with supports in \(Y\). Witt groups with supports have standard properties: excision, exact localization sequence, exact Mayer-Vietoris sequence and homotopy invariance. Moreover, in “good” cases (e.g. \(X,Y\) affine regular) if \(d=\text{codim} _{X}Y\), then there is a natural isomorphism \(W^{i-d}(Y)\cong W_{Y}^{i}(X)\). All this together form a flexible and effective tool for computing Witt groups of schemes.
In the paper under review authors show how this tool works for punctured affine space \({\mathbb U}_{X}^{n}={\mathbb A}_{X}^{n}\backslash pt\) over a regular scheme \(X\) of finite Krull dimension. There is a general construction of Koszul complex, which defines trivial element of Witt group of affine space. A ”half” of this complex defines a nontrivial element \( \varepsilon _{X}^{(n)}\) of \(W^{n-1}\left( {\mathbb U}_{X}^{n}\right) \) for \(n\geq 2\), which is locally hyperbolic and can not be extended to the affine space \({\mathbb A}_{X}^{n}\). Moreover, there are split exact sequences \[ 0\longrightarrow W^{i}(X)\longrightarrow W^{i}\left( {\mathbb U}_{X}^{n}\right) \longrightarrow W_{X}^{i}\left({\mathbb A}_{X}^{n}\right) \longrightarrow 0 \] which yield isomorphisms \[ W^{i}\left({\mathbb U}_{X}^{n}\right) \cong W^{i}(X)\oplus \varepsilon _{X}^{(n)}W^{i+1-n}(X) \] for all integers \(i\). With respect to the gradded multiplication \( \varepsilon _{X}^{(n)}\cdot \varepsilon _{X}^{(n)}=0\). As an application, the Witt groups of split affine quadric of odd dimension are computed - such a quadric is an affine bundle over a punctured affine space.


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