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Products of degenerate quadratic forms. (English) Zbl 1087.18008
The title of the paper could easily be quite provocative: “Products of degenerate quadratic forms need not be degenerate”. Of course, not in the classical theory of, say, finitely generated projective modules. However, in the author’s general framework of tensor triangulated categories with duality non-degeneracy of a product of forms depends on an invariant called consanguinity of the factors rather than on the non-degeneracy of the factors. Another surprising phenomenon is that the product of a degenerate form and a metabolic form can be a non-degenerate non-metabolic form. All this can be used to construct Witt classes of the product of two forms out of factors that do not produce Witt classes.
The setup follows the author’s two papers [P. Balmer, “Triangular Witt groups. I: The 12-term localization exact sequence”, K-Theory 19, 311–363 (2000; Zbl 0953.18003); “Triangular Witt groups. II: From usual to derived”, Math. Z. 236, 351–382 (2001; Zbl 1004.18010)]. The concept of product of forms comes from S. Gille and A. Nenashev [“Pairings in triangular Witt theory”, J. Algebra 261, 292–309 (2003; Zbl 1016.18007)]. The exposition is quite detailed with many enlightening comments. Besides the two main results mentioned above, a Leibniz-type formula is proved for the product with respect to the symmetric cone construction, and the total Witt group of regular projective space is computed.

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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