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Hermitian \(K\)-theory, derived equivalences and Karoubi’s fundamental theorem. (English) Zbl 1360.19008
In the paper under review, the author studies the problem of invariance under derived equivalences for the higher Grothendieck-Witt groups. Recall that by a classical result by R. W. Thomason and T. Trobaugh [Prog. Math. 88, None (1990; Zbl 0731.14001)] algebraic \(K\)-theory is invariant under derived equivalences. This, together with F. Waldhausen’s Fibration Theorem [Lect. Notes Math. 1126, 318–419 (1985; Zbl 0579.18006)] implies the Localization Theorem stating that a short exact sequence of triangulated categories induces a long exact sequence of algebraic \(K\)-groups. The higher Grothendieck-Witt groups were introduced by the author in his earlier paper [Invent. Math. 179, No. 2, 349–433 (2010; Zbl 1193.19005)], and, roughly speaking, are the algebraic analogue of the topological KO-groups, or of Atiyah’s KR-theory. In the paper under review the author shows that the higher Grothendieck-Witt groups are invariant under derived equivalences. This, together with the Fibration Theorem earlier proven by the author in [Invent. Math. 179, No. 2, 349–433 (2010; Zbl 1193.19005)], implies the analogues of the Thomason-Trobaugh-Waldhausen Localization Theorem. Numerous applications of this version of the Localization Theorem are provided. In particular, the recent solution of the Quillen-Lichtenbaum conjecture in Hermitian \(K\)-theory by A. J. Berrick et al. [Adv. Math. 278, 34–55 (2015; Zbl 1346.19006)], uses results now published in the paper.

19G38 Hermitian \(K\)-theory, relations with \(K\)-theory of rings
19G12 Witt groups of rings
18E30 Derived categories, triangulated categories (MSC2010)
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