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Blumberg, Andrew J.; Gepner, David; Tabuada, Gonçalo

\(K\)-theory of endomorphisms via noncommutative motives. (English) Zbl 1375.19006

Trans. Am. Math. Soc. 368, No. 2, 1435-1465 (2016).
Summary: We extend the \( K\)-theory of endomorphisms functor from ordinary rings to (stable) \( \infty \)-categories. We show that \( \mathrm{KEnd}(-)\) descends to the category of noncommutative motives, where it is corepresented by the noncommutative motive associated to the tensor algebra \( \mathbb{S}[t]\) of the sphere spectrum \( \mathbb{S}\). Using this corepresentability result, we classify all the natural transformations of \( \mathrm{KEnd}(-)\) in terms of an integer plus a fraction between polynomials with constant term \( 1\); this solves a problem raised by Almkvist in the seventies. Finally, making use of the multiplicative coalgebra structure of \( \mathbb{S}[t]\), we explain how the (rational) Witt vectors can also be recovered from the symmetric monoidal category of noncommutative motives. Along the way we show that the \( K_0\)-theory of endomorphisms of a connective ring spectrum \( R\) equals the \( K_0\)-theory of endomorphisms of the underlying ordinary ring \( \pi _0R\).

Cited in 2 Documents

MSC:

19D10 Algebraic \(K\)-theory of spaces
19D25 Karoubi-Villamayor-Gersten \(K\)-theory
19D55 \(K\)-theory and homology; cyclic homology and cohomology
18D20 Enriched categories (over closed or monoidal categories)
55N15 Topological \(K\)-theory

Keywords:

\(K\)-theory of endomorphisms; noncommutative motives; stable \(\infty\)-categories; Witt vectors
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\textit{A. J. Blumberg} et al., Trans. Am. Math. Soc. 368, No. 2, 1435--1465 (2016; Zbl 1375.19006)
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References:

[1] Almkvist, Gert, \(K\)-theory of endomorphisms, J. Algebra, 55, 2, 308-340 (1978) · Zbl 0414.18009
[2] Almkvist, Gert, The Grothendieck ring of the category of endomorphisms, J. Algebra, 28, 375-388 (1974) · Zbl 0281.18012
[3] Antieau, Benjamin; Gepner, David, Brauer groups and \'etale cohomology in derived algebraic geometry, Geom. Topol., 18, 2, 1149-1244 (2014) · Zbl 1308.14021
[4] [Barwick] C. Barwick, \newblock On the algebraic K-theory of higher categories. \newblock Preprint, arXiv:1204.3607.
[5] [Barwickexact] C. Barwick, \newblock On exact \(\i \)-categories and the Theorem of the Heart. \newblock Preprint, arXiv:1212.5232.
[6] [BarwickQ] C. Barwick, \newblock On the Q-construction for exact \(\i \)-categories. \newblock Preprint, arXiv:1301.4725.
[7] Betley, Stanislaw; Schlichtkrull, Christian, The cyclotomic trace and curves on \(K\)-theory, Topology, 44, 4, 845-874 (2005) · Zbl 1080.19005
[8] Blumberg, Andrew J.; Gepner, David; Tabuada, Gon{\c{c}}alo, A universal characterization of higher algebraic \(K\)-theory, Geom. Topol., 17, 2, 733-838 (2013) · Zbl 1267.19001
[9] Blumberg, Andrew J.; Gepner, David; Tabuada, Gon{\c{c}}alo, Uniqueness of the multiplicative cyclotomic trace, Adv. Math., 260, 191-232 (2014) · Zbl 1297.19002
[10] Bloch, Spencer, Algebraic \(K\)-theory and crystalline cohomology, Inst. Hautes \'Etudes Sci. Publ. Math., 47, 187-268 (1978) (1977) · Zbl 0388.14010
[11] Blumberg, Andrew J.; Mandell, Michael A., The localization sequence for the algebraic \(K\)-theory of topological \(K\)-theory, Acta Math., 200, 2, 155-179 (2008) · Zbl 1149.18008
[12] Blumberg, Andrew J.; Mandell, Michael A., Algebraic \(K\)-theory and abstract homotopy theory, Adv. Math., 226, 4, 3760-3812 (2011) · Zbl 1217.19003
[13] Blumberg, Andrew J.; Mandell, Michael A., Localization theorems in topological Hochschild homology and topological cyclic homology, Geom. Topol., 16, 2, 1053-1120 (2012) · Zbl 1282.19004
[14] Bondarko, M. V., Weight structures vs. \(t\)-structures; weight filtrations, spectral sequences, and complexes (for motives and in general), J. K-Theory, 6, 3, 387-504 (2010) · Zbl 1303.18019
[15] Dwyer, W. G.; Kan, D. M., Simplicial localizations of categories, J. Pure Appl. Algebra, 17, 3, 267-284 (1980) · Zbl 0485.18012
[16] Dwyer, W. G.; Kan, D. M., Calculating simplicial localizations, J. Pure Appl. Algebra, 18, 1, 17-35 (1980) · Zbl 0485.18013
[17] Dwyer, W. G.; Kan, D. M., Function complexes for diagrams of simplicial sets, Nederl. Akad. Wetensch. Indag. Math., 45, 2, 139-147 (1983) · Zbl 0524.55021
[18] Dwyer, William G.; Hirschhorn, Philip S.; Kan, Daniel M.; Smith, Jeffrey H., Homotopy limit functors on model categories and homotopical categories, Mathematical Surveys and Monographs 113, viii+181 pp. (2004), American Mathematical Society, Providence, RI · Zbl 1072.18012
[19] Elmendorf, A. D.; Kriz, I.; Mandell, M. A.; May, J. P., Rings, modules, and algebras in stable homotopy theory, with an appendix by M. Cole. Mathematical Surveys and Monographs 47, xii+249 pp. (1997), American Mathematical Society, Providence, RI · Zbl 0894.55001
[20] [Fiore] T. Fiore and W. Lueck, \newblockWaldhausen additivity: Classical and quasicategorical. \newblock Preprint, arXiv:1207.6613.
[21] [Glas] S. Glasman, Day convolution for \(\infty \)-categories, Preprint, arXiv:1308.4940. · Zbl 1375.18041
[22] Grayson, Daniel R., \(K_2\) and the \(K\)-theory of automorphisms, J. Algebra, 58, 1, 12-30 (1979) · Zbl 0413.18011
[23] Grayson, Daniel R., The \(K\)-theory of endomorphisms, J. Algebra, 48, 2, 439-446 (1977) · Zbl 0413.18010
[24] Grayson, Daniel R., Grothendieck rings and Witt vectors, Comm. Algebra, 6, 3, 249-255 (1978) · Zbl 0383.13009
[25] [SGA6] A. Grothendieck et al., \newblock SGA 6.
[26] Hazewinkel, Michiel, Operations in the \(K\)-theory of endomorphisms, J. Algebra, 84, 2, 285-304 (1983) · Zbl 0534.13010
[27] Hazewinkel, Michiel, Witt vectors. I. Handbook of algebra. Vol. 6, Handb. Algebr. 6, 319-472 (2009), Elsevier/North-Holland, Amsterdam · Zbl 1221.13036
[28] Hesselholt, Lars; Madsen, Ib, Cyclic polytopes and the \(K\)-theory of truncated polynomial algebras, Invent. Math., 130, 1, 73-97 (1997) · Zbl 0884.19004
[29] Hesselholt, Lars, On the \(p\)-typical curves in Quillen’s \(K\)-theory, Acta Math., 177, 1, 1-53 (1996) · Zbl 0892.19003
[30] Hovey, Mark, Model categories, Mathematical Surveys and Monographs 63, xii+209 pp. (1999), American Mathematical Society, Providence, RI · Zbl 0909.55001
[31] Lurie, Jacob, Higher topos theory, Annals of Mathematics Studies 170, xviii+925 pp. (2009), Princeton University Press, Princeton, NJ · Zbl 1175.18001
[32] [HA] J. Lurie, \newblockHigher algebra. \newblock Preprint, available at www.math.harvard.edu/lurie.
[33] Lindenstrauss, Ayelet; McCarthy, Randy, On the algebraic K-theory of formal power series, J. K-Theory, 10, 1, 165-189 (2012) · Zbl 1257.19002
[34] Mandell, M. A.; Shipley, B., A telescope comparison lemma for THH, Topology Appl., 117, 2, 161-174 (2002) · Zbl 1098.55501
[35] Neeman, Amnon, Triangulated categories, Annals of Mathematics Studies 148, viii+449 pp. (2001), Princeton University Press, Princeton, NJ · Zbl 0974.18008
[36] Quillen, Daniel, Higher algebraic \(K\)-theory. I. Algebraic \(K\)-theory, I: Higher \(K\)-theories, Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972, 85-147 (1973), Lecture Notes in Math., Vol. 341, Springer, Berlin · Zbl 0292.18004
[37] [Schwede] S. Schwede, \newblockSymmetric spectra. \newblock Preprint, http://www.math.uni-bonn.de/ schwede/.
[38] Schwede, Stefan; Shipley, Brooke, Stable model categories are categories of modules, Topology, 42, 1, 103-153 (2003) · Zbl 1013.55005
[39] Stienstra, Jan, Cartier-Dieudonn\'e theory for Chow groups, J. Reine Angew. Math., 355, 1-66 (1985) · Zbl 0545.14013
[40] Stienstra, Jan, Operations in the higher \(K\)-theory of endomorphisms. Current trends in algebraic topology, Part 1, London, Ont., 1981, CMS Conf. Proc. 2, 59-115 (1982), Amer. Math. Soc., Providence, R.I.
[41] Tabuada, Gon{\c{c}}alo, Homotopy theory of spectral categories, Adv. Math., 221, 4, 1122-1143 (2009) · Zbl 1168.55008
[42] Thomason, R. W.; Trobaugh, Thomas, Higher algebraic \(K\)-theory of schemes and of derived categories. The Grothendieck Festschrift, Vol. III, Progr. Math. 88, 247-435 (1990), Birkh\"auser Boston, Boston, MA
[43] Waldhausen, Friedhelm, Algebraic \(K\)-theory of spaces. Algebraic and geometric topology, New Brunswick, N.J., 1983, Lecture Notes in Math. 1126, 318-419 (1985), Springer, Berlin
[44] [Witt] E. Witt. \newblock Theorie der quadratischen Formen in beliebigen K\"orpern. \newblock J. Reine Angew. Math. 176 (1937), 31-44. · Zbl 0015.05701
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