# zbMATH — the first resource for mathematics

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.

##### MSC:
 19G38 Hermitian $$K$$-theory, relations with $$K$$-theory of rings 19G12 Witt groups of rings 18E30 Derived categories, triangulated categories (MSC2010)
Full Text:
##### References:
 [1] Adem, A.; Cohen, R. L.; Dwyer, W. G., Generalized Tate homology, homotopy fixed points and the transfer, (Algebraic Topology, Evanston, IL, 1988, Contemp. Math., vol. 96, (1989), Amer. Math. Soc. Providence, RI), 1-13 [2] Asok, Aravind; Fasel, Jean, A cohomological classification of vector bundles on smooth affine threefolds, Duke Math. J., 163, 14, 2561-2601, (2014) [3] Balmer, Paul, Triangular Witt groups, I: the 12-term localization exact sequence, K-Theory, 19, 4, 311-363, (2000) [4] Balmer, Paul, Triangular Witt groups, II: from usual to derived, Math. Z., 236, 2, 351-382, (2001) [5] Balmer, Paul, Witt cohomology, Mayer-Vietoris, homotopy invariance and the Gersten conjecture, K-Theory, 23, 1, 15-30, (2001) [6] Bousfield, A. K.; Friedlander, E. M., Homotopy theory of γ-spaces, spectra, and bisimplicial sets, (Geometric Applications of Homotopy Theory, II, Proc. Conf., Evanston, Ill., 1977, Lecture Notes in Math., vol. 658, (1978), Springer Berlin), 80-130 [7] Bondal, A. I.; Kapranov, M. M., Enhanced triangulated categories, Mat. Sb., 181, 5, 669-683, (1990) [8] Berrick, A. J.; Karoubi, M., Hermitian K-theory of the integers, Am. J. Math., 127, 4, 785-823, (2005) [9] Berrick, A. J.; Karoubi, M.; Schlichting, M.; Østvær, P. A., The homotopy fixed point theorem and the Quillen-lichtenbaum conjecture in Hermitian K-theory, Adv. Math., 278, 34-55, (2015) [10] Brown, Kenneth S., Cohomology of groups, Graduate Texts in Mathematics, vol. 87, (1994), Springer-Verlag New York, corrected reprint of the 1982 original [11] Balmer, Paul; Schlichting, Marco, Idempotent completion of triangulated categories, J. Algebra, 236, 2, 819-834, (2001) [12] Cortiñas, G.; Haesemeyer, C.; Schlichting, M.; Weibel, C., Cyclic homology, cdh-cohomology and negative K-theory, Ann. Math. (2), 167, 2, 549-573, (2008) [13] Cortiñas, G.; Haesemeyer, C.; Weibel, C., K-regularity, cdh-fibrant Hochschild homology, and a conjecture of vorst, J. Am. Math. Soc., 21, 2, 547-561, (2008) [14] Drinfeld, Vladimir, DG quotients of DG categories, J. Algebra, 272, 2, 643-691, (2004) [15] Fasel, J.; Rao, R. A.; Swan, R. G., On stably free modules over affine algebras, Publ. Math. Inst. Hautes Études Sci., 116, 223-243, (2012) [16] Friedlander, Eric M.; Suslin, Andrei, The spectral sequence relating algebraic K-theory to motivic cohomology, Ann. Sci. Éc. Norm. Supér., 35, 6, 773-875, (2002) [17] Fasel, J.; Srinivas, V., Chow-Witt groups and Grothendieck-Witt groups of regular schemes, Adv. Math., 221, 1, 302-329, (2009) [18] Gersten, S. M., On the spectrum of algebraic K-theory, Bull. Am. Math. Soc., 78, 216-219, (1972) [19] Geisser, Thomas; Hesselholt, Lars, Topological cyclic homology of schemes, (Algebraic K-Theory, Seattle, WA, 1997, Proc. Sympos. Pure Math., vol. 67, (1999), Amer. Math. Soc. Providence, RI), 41-87 [20] Gille, Stefan, The general dévissage theorem for Witt groups of schemes, Arch. Math. (Basel), 88, 4, 333-343, (2007) [21] Grayson, Daniel, Higher algebraic K-theory, II (after daniel Quillen), (Algebraic K-Theory, Proc. Conf., Northwestern Univ., Evanston, Ill., 1976, Lecture Notes in Math., vol. 551, (1976), Springer Berlin), 217-240 [22] Greenlees, J. P.C., Tate cohomology in axiomatic stable homotopy theory, (Cohomological Methods in Homotopy Theory, Bellaterra, 1998, Prog. Math., vol. 196, (2001), Birkhäuser Basel), 149-176 [23] Happel, Dieter, On the derived category of a finite-dimensional algebra, Comment. Math. Helv., 62, 3, 339-389, (1987) [24] Hartshorne, Robin, Residues and duality, Lecture Notes in Mathematics, vol. 20, (1966), Springer-Verlag Berlin, Lecture Notes of a Seminar on the Work of A. Grothendieck, Given at Harvard 1963/64, with an Appendix by P. Deligne [25] Hirschhorn, Philip S., Model categories and their localizations, Mathematical Surveys and Monographs, vol. 99, (2003), American Mathematical Society Providence, RI [26] Hornbostel, Jens, $$A^1$$-representability of Hermitian K-theory and Witt groups, Topology, 44, 3, 661-687, (2005) [27] Hovey, Mark, Model categories, Mathematical Surveys and Monographs, vol. 63, (1999), American Mathematical Society Providence, RI [28] Hovey, Mark, Spectra and symmetric spectra in general model categories, J. Pure Appl. Algebra, 165, 1, 63-127, (2001) [29] Hovey, Mark; Shipley, Brooke; Smith, Jeff, Symmetric spectra, J. Am. Math. Soc., 13, 1, 149-208, (2000) [30] Jardine, J. F., Generalized étale cohomology theories, Prog. Math., vol. 146, (1997), Birkhäuser Verlag Basel [31] Karoubi, Max, Foncteurs dérivés et K-théorie, (Séminaire Heidelberg-Saarbrücken-Strasbourg sur la K-théorie (1967/68), Lecture Notes in Mathematics, vol. 136, (1970), Springer Berlin), 107-186 [32] Karoubi, Max, Périodicité de la K-théorie hermitienne, (Algebraic K-Theory, III: Hermitian K-Theory and Geometric Applications, Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972, Lecture Notes in Math., vol. 343, (1973), Springer Berlin), 301-411 [33] Karoubi, Max, Le théorème fondamental de la K-théorie hermitienne, Ann. Math. (2), 112, 2, 259-282, (1980) [34] Karoubi, Max, Stabilization of the Witt group, C. R. Math. Acad. Sci. Paris, 342, 3, 165-168, (2006) [35] Kelly, Gregory Maxwell, Basic concepts of enriched category theory, London Mathematical Society Lecture Note Series, vol. 64, (1982), Cambridge University Press Cambridge [36] Keller, Bernhard, Chain complexes and stable categories, Manuscr. Math., 67, 4, 379-417, (1990) [37] Keller, Bernhard, Derived categories and their uses, (Handbook of Algebra, vol. 1, (1996), North-Holland Amsterdam), 671-701 [38] Keller, Bernhard, On the cyclic homology of exact categories, J. Pure Appl. Algebra, 136, 1, 1-56, (1999) [39] Keller, Bernhard, On differential graded categories, (International Congress of Mathematicians, vol. II, (2006), Eur. Math. Soc. Zürich), 151-190 [40] Kelly, G. M.; Mac Lane, S., Coherence in closed categories, J. Pure Appl. Algebra, 1, 1, 97-140, (1971) [41] Knebusch, Manfred, Symmetric bilinear forms over algebraic varieties, (Conference on Quadratic Forms—1976, Proc. Conf., Queen’s Univ., Kingston, Ont., 1976, Queen’s Papers in Pure and Appl. Math., vol. 46, (1977), Queen’s Univ. Kingston, Ont.), 103-283 [42] Kobal, Damjan, K-theory, Hermitian K-theory and the karoubi tower, K-Theory, 17, 2, 113-140, (1999) [43] Karoubi, Max; Schlichting, Marco; Weibel, Charles, The Witt group of real algebraic varieties, J. Topol., 9, 4, 1257-1302, (2016) [44] May, J. P., $$E_\infty$$ spaces, group completions, and permutative categories, (New Developments in Topology, Proc. Sympos. Algebraic Topology, Oxford, 1972, London Math. Soc. Lecture Note Ser., vol. 11, (1974), Cambridge Univ. Press London), 61-93 [45] May, J. Peter, Classifying spaces and fibrations, Mem. Am. Math. Soc., 1, 1, 155, (1975) [46] McCord, M. C., Classifying spaces and infinite symmetric products, Trans. Am. Math. Soc., 146, 273-298, (1969) [47] Milnor, John; Husemoller, Dale, Symmetric bilinear forms, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 73, (1973), Springer-Verlag New York [48] Mandell, M. A.; May, J. P.; Schwede, S.; Shipley, B., Model categories of diagram spectra, Proc. Lond. Math. Soc. (3), 82, 2, 441-512, (2001) [49] Morel, Fabian, $$\mathbb{A}^1$$-algebraic topology over a field, Lecture Notes in Mathematics, vol. 2052, (2012), Springer-Verlag Berlin [50] Morel, Fabien; Voevodsky, Vladimir, $$\mathbf{A}^1$$-homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math., 90, 45-143, (2001), 1999 [51] Neeman, Amnon, The derived category of an exact category, J. Algebra, 135, 2, 388-394, (1990) [52] Panin, Ivan; Walter, Charles, On the motivic commutative ring spectrum BO, (2010) [53] Quillen, Daniel, Higher algebraic K-theory, I, (Algebraic K-Theory, I: Higher K-Theories, Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972, Lecture Notes in Math., vol. 341, (1973), Springer Berlin), 85-147 [54] Schwede, Stefan, Symmetric spectra [55] Schlichting, Marco, Hermitian K-theory on a theorem of giffen, K-Theory, 32, 3, 253-267, (2004) [56] Schlichting, Marco, Negative K-theory of derived categories, Math. Z., 253, 1, 97-134, (2006) [57] Schlichting, Marco, Hermitian K-theory of exact categories, J. K-Theory, 5, 1, 105-165, (2010) [58] Schlichting, Marco, The Mayer-Vietoris principle for Grothendieck-Witt groups of schemes, Invent. Math., 179, 2, 349-433, (2010) [59] Schlichting, Marco, Higher algebraic K-theory, (Topics in Algebraic and Topological K-Theory, Lecture Notes in Math., vol. 2008, (2011), Springer Berlin), 167-241 [60] Segal, Graeme, Categories and cohomology theories, Topology, 13, 293-312, (1974) [61] Swan, Richard G., Vector bundles and projective modules, Trans. Am. Math. Soc., 105, 264-277, (1962) [62] Swan, Richard G., Topological examples of projective modules, Trans. Am. Math. Soc., 230, 201-234, (1977) [63] Thomason, R. W., LES K-groupes d’un fibré projectif, (Algebraic K-Theory and Algebraic Topology, Lake Louise, AB, 1991, NATO Adv. Stud. Inst. Ser., Ser. C, Math. Phys. Sci., vol. 407, (1993), Kluwer Acad. Publ. Dordrecht), 243-248 [64] Thomason, R. W., LES K-groupes d’un schéma éclaté et une formule d’intersection excédentaire, Invent. Math., 112, 1, 195-215, (1993) [65] Thomason, R. W., The classification of triangulated subcategories, Compos. Math., 105, 1, 1-27, (1997) [66] Thomason, R. W.; Trobaugh, Thomas, Higher algebraic K-theory of schemes and of derived categories, (The Grothendieck Festschrift, vol. III, Prog. Math., vol. 88, (1990), Birkhäuser Boston Boston, MA), 247-435 [67] Wagoner, J. B., Delooping classifying spaces in algebraic K-theory, Topology, 11, 349-370, (1972) [68] Waldhausen, Friedhelm, Algebraic K-theory of generalized free products, I, II, Ann. Math. (2), 108, 1, 135-204, (1978) [69] Waldhausen, Friedhelm, Algebraic K-theory of spaces, (Algebraic and Geometric Topology, New Brunswick, NJ, 1983, Lecture Notes in Math., vol. 1126, (1985), Springer Berlin), 318-419 [70] Charles Walter, Grothendieck-Witt groups of projective bundles, K-theory preprint archive, 2003. [71] Charles Walter, Grothendieck-Witt groups of triangulated categories, K-theory preprint archive, 2003. [72] Weibel, Charles A., K-theory of Azumaya algebras, Proc. Am. Math. Soc., 81, 1, 1-7, (1981) [73] Weiss, Michael; Williams, Bruce, Automorphisms of manifolds and algebraic K-theory, II, J. Pure Appl. Algebra, 62, 1, 47-107, (1989) [74] Weiss, Michael S.; Williams, Bruce, Products and duality in waldhausen categories, Trans. Am. Math. Soc., 352, 2, 689-709, (2000) [75] Zibrowius, Marcus, Witt groups of complex cellular varieties, Doc. Math., 16, 465-511, (2011)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.