Hoyois, Marc

Cdh Descent in Equivariant Homotopy \(K\)-Theory

Doc. Math. 25, 457-482 (2020)
DOI: 10.25537/dm.2020v25.457-482
Communicated by Mike Hill


We construct geometric models for classifying spaces of linear algebraic groups in \(G\)-equivariant motivic homotopy theory, where \(G\) is a tame group scheme. As a consequence, we show that the equivariant motivic spectrum representing the homotopy \(K\)-theory of \(G\)-schemes (which we construct as an \(E_\infty \)-ring) is stable under arbitrary base change, and we deduce that the homotopy \(K\)-theory of \(G\)-schemes satisfies cdh descent.

Mathematics Subject Classification

14F42, 14D23, 19D25, 14A20


algebraic \(K\)-theory, algebraic stacks


  • [AOV08]. D. Abramovich, M. Olsson, and A. Vistoli, Tame stacks in positive characteristic, Ann. I. Fourier 58 (2008), no. 4, pp. 1057-1091. DOI 10.5802/aif.2378; zbl 1222.14004; MR2427954; arxiv math/0703310.
  • [CHSW08]. G. Cortiñas, C. Haesemeyer, M. Schlichting, and C. A. Weibel, Cyclic homology, cdh-cohomology and negative K-theory, Ann. Math. 167 (2008), no. 2, pp. 549-573. DOI 10.4007/annals.2008.167.549; zbl 1191.19003; MR2415380; arxiv math/0502255.
  • [Cis13]. D.-C. Cisinski, Descente par éclatements en \(K\)-théorie invariante par homotopie, Ann. Math. 177 (2013), no. 2, pp. 425-448. DOI 10.4007/annals.2013.177.2.2; zbl 1264.19003; MR3010804; arxiv 1003.1487.
  • [Gro17]. P. Gross, Tensor generators on schemes and stacks, Algebr. Geom. 4 (2017), no. 4, pp. 501-522. DOI 10.14231/AG-2017-026; zbl 1412.14002; MR3683505; arxiv 1306.5418.
  • [HK19]. M. Hoyois and A. Krishna, Vanishing theorems for the negative \(K\)-theory of stacks, Ann. K-Theory 4 (2019), no. 3, pp. 439-472. DOI 10.2140/akt.2019.4.439; zbl 07146016; MR4043465; arxiv 1705.02295.
  • [HR15]. J. Hall and D. Rydh, Algebraic groups and compact generation of their derived categories of representations, Indiana Univ. Math. J. 64 (2015), no. 6, pp. 1903-1923. DOI 10.1512/iumj.2015.64.5719; zbl 1348.14045; MR3436239; arxiv 1405.1890.
  • [Hae04]. C. Haesemeyer, Descent properties of homotopy \(K\)-theory, Duke Math. J. 125 (2004), no. 3, pp. 589-620. DOI 10.1215/S0012-7094-04-12534-5; zbl 1079.19001; MR2166754.
  • [Hoy17]. M. Hoyois, The six operations in equivariant motivic homotopy theory, Adv. Math. 305 (2017), pp. 197-279. DOI 10.1016/j.aim.2016.09.031; zbl 1400.14065; MR3570135; arxiv 1509.02145.
  • [KØ12]. A. Krishna and P. A. \Ostvær, Nisnevich descent for \(K\)-theory of Deligne-Mumford stacks, J. K-Theory 9 (2012), no. 2, pp. 291-331. DOI 10.1017/is011006028jkt161; zbl 1284.19005; MR2922391; arxiv 1005.0968.
  • [KR18]. A. Krishna and C. Ravi, Algebraic \(K\)-theory of quotient stacks, Ann. K-Theory 3 (2018), no. 2, pp. 207-233. DOI 10.2140/akt.2018.3.207; MR3781427; zbl 1423.19007; arxiv 1509.05147.
  • [KS17]. M. Kerz and F. Strunk, On the vanishing of negative homotopy \(K\)-theory, J. Pure Appl. Alg. 221 (2017), no. 7, pp. 1641-1644. DOI 10.1016/j.jpaa.2016.12.021; zbl 1372.19003; MR3614971; arxiv 1601.08075.
  • [KST18]. M. Kerz, F. Strunk, and G. Tamme, Algebraic \(K\)-theory and descent for blow-ups, Invent. Math. 211 (2018), no. 2, pp. 523-577. DOI 10.1007/s00222-017-0752-2; zbl 1391.19007; MR3748313; arxiv 1611.08466.
  • [Kel14]. S. Kelly, Vanishing of negative \(K\)-theory in positive characteristic, Compos. Math. 150 (2014), pp. 1425-1434. DOI 10.1112/S0010437X14007301; zbl 1301.19001; MR3252025; arxiv 1112.5206.
  • [Lur09]. J. Lurie, Higher Topos Theory, Annals of Mathematical Studies, vol. 170, Princeton University Press, 2009. DOI 10.1515/9781400830558; zbl 1175.18001; MR2522659; arxiv math/0608040.
  • [Lur17]. J. Lurie, Higher Algebra, September 2017. http://www.math.harvard.edu/~lurie/papers/HA.pdf.
  • [Lur18]. J. Lurie, Spectral Algebraic Geometry, February 2018. http://www.math.harvard.edu/~lurie/papers/SAG-rootfile.pdf.
  • [MV99]. F. Morel and V. Voevodsky, \( \mathbb{A}^1\)-homotopy theory of schemes, Publ. Math. I.H.É.S. 90 (1999), pp. 45-143. DOI 10.1007/BF02698831; zbl 0983.14007; MR1813224.
  • [Rob15]. M. Robalo, \(K\)-theory and the bridge from motives to noncommutative motives, Adv. Math. 269 (2015), pp. 399-550. DOI 10.1016/j.aim.2014.10.011; zbl 1315.14030; MR3281141; arxiv 1306.3795.
  • [Ryd13]. D. Rydh, Existence and properties of geometric quotients, J. Algebraic Geom. 22 (2013), pp. 629-669. DOI 10.1090/S1056-3911-2013-00615-3; zbl 1278.14003; MR3084720; arxiv 0708.3333.
  • [TT90]. R. W. Thomason and T. Trobaugh, Higher algebraic K-theory of schemes and of derived categories, The Grothendieck Festschrift III, Progress in Mathematics, vol. 88, Birkhäuser, 1990, pp. 247-435. zbl 0731.14001; MR1106918.
  • [Tho87]. R. W. Thomason, Algebraic K-theory of group scheme actions, Algebraic Topology and Algebraic K-theory (W. Browder, ed.), Annals of Mathematical Studies, vol. 113, Princeton University Press, 1987. zbl 0701.19002; MR0921490.
  • [Wei89]. C. A. Weibel, Homotopy algebraic \(K\)-theory, Algebraic K-Theory and Number Theory, Contemp. Math., vol. 83, AMS, 1989, pp. 461-488. zbl 0669.18007; MR0991991.


Hoyois, Marc
Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany