Examples of Descent up to Nilpotence

  • Akhil MathewEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 242)


We give a survey of the ideas of descent and nilpotence, beginning with the theory of thick subcategories. We focus on examples arising from chromatic homotopy theory (such as Rognes’ Galois extensions) and from group actions, as well as a few examples in algebra. These ideas provide tools for studying certain invariants of tensor-triangulated categories.



I would like to thank Bhargav Bhatt, Srikanth Iyengar, and Jacob Lurie for helpful discussions. I would especially like to thank my collaborators Niko Naumann and Justin Noel; much of this material is drawn from [54, 55]. Most of all, I would like to thank Mike Hopkins: most of these ideas originated in his work. I am grateful to the referee and to Niko Naumann for several corrections. While this article was written, I was supported by the NSF Graduate Fellowship under grant DGE-114415.


  1. 1.
    Revêtements étales et groupe fondamental (SGA 1). Documents Mathématiques (Paris) [Mathematical Documents (Paris)], 3. Société Mathématique de France, Paris, 2003. Séminaire de géométrie algébrique du Bois Marie 1960–61. [Algebraic Geometry Seminar of Bois Marie 1960-61], Directed by A. Grothendieck, With two papers by M. Raynaud, Updated and annotated reprint of the 1971 original [Lecture Notes in Math., 224, Springer, Berlin; MR0354651 (50 #7129)].Google Scholar
  2. 2.
    M. Ando, A. J. Blumberg, D. Gepner, M. J. Hopkins, and C. Rezk. An \(\infty \)-categorical approach to \(R\)-line bundles, \(R\)-module Thom spectra, and twisted \(R\)-homology. J. Topol., 7(3):869–893, 2014.MathSciNetzbMATHGoogle Scholar
  3. 3.
    L. L. Avramov, R.-O. Buchweitz, S. B. Iyengar, and C. Miller. Homology of perfect complexes. Adv. Math., 223(5):1731–1781, 2010.MathSciNetzbMATHGoogle Scholar
  4. 4.
    A. Baker and B. Richter. Invertible modules for commutative \({\mathbb{S}}\)-algebras with residue fields. Manuscripta Math., 118(1):99–119, 2005.MathSciNetzbMATHGoogle Scholar
  5. 5.
    P. Balmer. The spectrum of prime ideals in tensor triangulated categories. J. Reine Angew. Math., 588:149–168, 2005.MathSciNetzbMATHGoogle Scholar
  6. 6.
    P. Balmer. Descent in triangulated categories. Math. Ann., 353(1):109–125, 2012.MathSciNetzbMATHGoogle Scholar
  7. 7.
    P. Balmer. Separable extensions in tensor-triangular geometry and generalized Quillen stratification. Ann. Sci. Éc. Norm. Supér. (4), 49(4):907–925, 2016.MathSciNetzbMATHGoogle Scholar
  8. 8.
    P. Balmer, I. Dell’Ambrogio, and B. Sanders. Restriction to finite-index subgroups as étale extensions in topology, KK-theory and geometry. Algebr. Geom. Topol., 15(5):3025–3047, 2015.zbMATHGoogle Scholar
  9. 9.
    R. Banerjee. Galois descent for real spectra. arXiv preprint arXiv:1305.4360, 2013.
  10. 10.
    T. Bauer. Convergence of the Eilenberg-Moore spectral sequence for generalized cohomology theories. 2008. Available at
  11. 11.
    B. Bhatt and P. Scholze. Projectivity of the Witt vector affine Grassmannian. Invent. Math., 209(2):329–423, 2017.MathSciNetzbMATHGoogle Scholar
  12. 12.
    A. K. Bousfield. The localization of spectra with respect to homology. Topology, 18(4):257–281, 1979.MathSciNetzbMATHGoogle Scholar
  13. 13.
    A. K. Bousfield and D. M. Kan. Homotopy limits, completions and localizations. Lecture Notes in Mathematics, Vol. 304. Springer-Verlag, Berlin-New York, 1972.Google Scholar
  14. 14.
    J. F. Carlson. Cohomology and induction from elementary abelian subgroups. Q. J. Math., 51(2):169–181, 2000.MathSciNetzbMATHGoogle Scholar
  15. 15.
    J. D. Christensen. Ideals in triangulated categories: phantoms, ghosts and skeleta. Adv. Math., 136(2):284–339, 1998.MathSciNetzbMATHGoogle Scholar
  16. 16.
    J. D. Christensen and N. P. Strickland. Phantom maps and homology theories. Topology, 37(2):339–364, 1998.MathSciNetzbMATHGoogle Scholar
  17. 17.
    D. Clausen, A. Mathew, N. Naumann, and J. Noel. Descent in algebraic \(K\)-theory and a conjecture of Ausoni-Rognes. arXiv preprint arXiv:1606.03328, 2016.
  18. 18.
    E. S. Devinatz, M. J. Hopkins, and J. H. Smith. Nilpotence and stable homotopy theory. I. Ann. of Math. (2), 128(2):207–241, 1988.MathSciNetzbMATHGoogle Scholar
  19. 19.
    W. G. Dwyer, J. P. C. Greenlees, and S. Iyengar. Duality in algebra and topology. Adv. Math., 200(2):357–402, 2006.MathSciNetzbMATHGoogle Scholar
  20. 20.
    A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May. Rings, modules, and algebras in stable homotopy theory, volume 47 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1997. With an appendix by M. Cole.Google Scholar
  21. 21.
    D. Gepner and T. Lawson. Brauer groups and Galois cohomology of commutative \({\mathbb{S}}\)-algebras. arXiv preprint arXiv:1607.01118.
  22. 22.
    J. González. A vanishing line in the BP\(\langle 1\rangle \)-Adams spectral sequence. Topology, 39(6):1137–1153, 2000.MathSciNetzbMATHGoogle Scholar
  23. 23.
    D. J. Green. The essential ideal in group cohomology does not square to zero. J. Pure Appl. Algebra, 193(1-3):129–139, 2004.MathSciNetzbMATHGoogle Scholar
  24. 24.
    J. P. C. Greenlees and J. P. May. Generalized Tate cohomology. Mem. Amer. Math. Soc., 113(543):viii+178, 1995.MathSciNetzbMATHGoogle Scholar
  25. 25.
    J. P. C. Greenlees and H. Sadofsky. The Tate spectrum of \(v_n\)-periodic complex oriented theories. Math. Z., 222(3):391–405, 1996.MathSciNetzbMATHGoogle Scholar
  26. 26.
    L. Gruson and C. U. Jensen. Dimensions cohomologiques reliées aux foncteurs \(\varprojlim ^{(i)}\). In Paul Dubreil and Marie-Paule Malliavin Algebra Seminar, 33rd Year (Paris, 1980), volume 867 of Lecture Notes in Math., pages 234–294. Springer, Berlin-New York, 1981.Google Scholar
  27. 27.
    D. Heard, A. Mathew, and V. Stojanoska. Picard groups of higher real \(K\)-theory spectra at height \(p-1\). Compos. Math., 153(9):1820–1854, 2017.MathSciNetzbMATHGoogle Scholar
  28. 28.
    K. Hess. Homotopic Hopf-Galois extensions: foundations and examples. In New topological contexts for Galois theory and algebraic geometry (BIRS 2008), volume 16 of Geom. Topol. Monogr., pages 79–132. Geom. Topol. Publ., Coventry, 2009.Google Scholar
  29. 29.
    K. Hess. A general framework for homotopic descent and codescent. arXiv preprint arXiv:1001.1556.
  30. 30.
    M. A. Hill, M. J. Hopkins, and D. C. Ravenel. On the nonexistence of elements of Kervaire invariant one. Ann. of Math. (2), 184(1):1–262, 2016.MathSciNetzbMATHGoogle Scholar
  31. 31.
    M. Hopkins and J. Lurie. Ambidexterity in \(K(n)\)-local stable homotopy theory. 2013. Available at
  32. 32.
    M. J. Hopkins. The mathematical work of Douglas C. Ravenel. Homology, Homotopy Appl., 10(3):1–13, 2008.MathSciNetzbMATHGoogle Scholar
  33. 33.
    M. J. Hopkins, N. J. Kuhn, and D. C. Ravenel. Generalized group characters and complex oriented cohomology theories. J. Amer. Math. Soc., 13(3):553–594 (electronic), 2000.MathSciNetGoogle Scholar
  34. 34.
    M. J. Hopkins, J. H. Palmieri, and J. H. Smith. Vanishing lines in generalized Adams spectral sequences are generic. Geom. Topol., 3:155–165 (electronic), 1999.MathSciNetzbMATHGoogle Scholar
  35. 35.
    M. J. Hopkins and J. H. Smith. Nilpotence and stable homotopy theory. II. Ann. of Math. (2), 148(1):1–49, 1998.MathSciNetzbMATHGoogle Scholar
  36. 36.
    M. Hovey and N. P. Strickland. Morava \(K\)-theories and localisation. Mem. Amer. Math. Soc., 139(666):viii+100, 1999.Google Scholar
  37. 37.
    J. R. Klein. Axioms for generalized Farrell-Tate cohomology. J. Pure Appl. Algebra, 172(2-3):225–238, 2002.MathSciNetzbMATHGoogle Scholar
  38. 38.
    N. J. Kuhn. Tate cohomology and periodic localization of polynomial functors. Invent. Math., 157(2):345–370, 2004.MathSciNetzbMATHGoogle Scholar
  39. 39.
    T. Lawson and N. Naumann. Commutativity conditions for truncated Brown-Peterson spectra of height 2. J. Topol., 5(1):137–168, 2012.MathSciNetzbMATHGoogle Scholar
  40. 40.
    J. Lurie. Higher topos theory, volume 170 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2009.Google Scholar
  41. 41.
    J. Lurie. Chromatic homotopy theory, 2010. Course notes available at
  42. 42.
    J. Lurie. DAG XIII: Rational and \(p\)-adic homotopy theory. 2011. Available at
  43. 43.
    J. Lurie. Derived algebraic geometry VII: spectral schemes. 2011. Available at
  44. 44.
    J. Lurie. Higher algebra. 2016. Available at
  45. 45.
    J. Lurie. Spectral algebraic geometry. Available at, 2016.
  46. 46.
    M. Mahowald. \(b\)o-resolutions. Pacific J. Math., 92(2):365–383, 1981.MathSciNetzbMATHGoogle Scholar
  47. 47.
    T. Marx. The restriction map in cohomology of finite \(2\)-groups. J. Pure Appl. Algebra, 67(1):33–37, 1990.MathSciNetzbMATHGoogle Scholar
  48. 48.
    A. Mathew. A thick subcategory theorem for modules over certain ring spectra. Geom. Topol., 19(4):2359–2392, 2015.MathSciNetzbMATHGoogle Scholar
  49. 49.
    A. Mathew. Torus actions on stable module categories, Picard groups, and localizing subcategories. arXiv preprint arXiv:1512.01716, 2015.
  50. 50.
    A. Mathew. The Galois group of a stable homotopy theory. Adv. Math., 291:403–541, 2016.MathSciNetzbMATHGoogle Scholar
  51. 51.
    A. Mathew. The homology of tmf. Homology Homotopy Appl., 18(2):1–29, 2016.MathSciNetzbMATHGoogle Scholar
  52. 52.
    A. Mathew. Torsion exponents in stable homotopy and the Hurewicz homomorphism. Algebr. Geom. Topol., 16(2):1025–1041, 2016.MathSciNetzbMATHGoogle Scholar
  53. 53.
    A. Mathew and L. Meier. Affineness and chromatic homotopy theory. J. Topol., 8(2):476–528, 2015.MathSciNetzbMATHGoogle Scholar
  54. 54.
    A. Mathew, N. Naumann, and J. Noel. Derived induction and restriction theory. 2015.Google Scholar
  55. 55.
    A. Mathew, N. Naumann, and J. Noel. Nilpotence and descent in equivariant stable homotopy theory. Adv. Math., 305:994–1084, 2017.MathSciNetzbMATHGoogle Scholar
  56. 56.
    A. Mathew and V. Stojanoska. The Picard group of topological modular forms via descent theory. Geom. Topol., 20(6):3133–3217, 2016.MathSciNetzbMATHGoogle Scholar
  57. 57.
    L. Meier. United elliptic homology. PhD thesis, University of Bonn, 2012.Google Scholar
  58. 58.
    F. Muro and O. Raventós. Transfinite Adams representability. Adv. Math., 292:111–180, 2016.MathSciNetzbMATHGoogle Scholar
  59. 59.
    A. Neeman. On a theorem of Brown and Adams. Topology, 36(3):619–645, 1997.MathSciNetzbMATHGoogle Scholar
  60. 60.
    D. Quillen. The spectrum of an equivariant cohomology ring. I, II. Ann. of Math. (2), 94:549–572; ibid. (2) 94 (1971), 573–602, 1971.Google Scholar
  61. 61.
    D. C. Ravenel. Nilpotence and periodicity in stable homotopy theory, volume 128 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1992. Appendix C by Jeff Smith.Google Scholar
  62. 62.
    C. Rezk. Notes on the Hopkins-Miller theorem. In Homotopy theory via algebraic geometry and group representations (Evanston, IL, 1997), volume 220 of Contemp. Math., pages 313–366. Amer. Math. Soc., Providence, RI, 1998.Google Scholar
  63. 63.
    E. Riehl and D. Verity. Homotopy coherent adjunctions and the formal theory of monads. Adv. Math., 286:802–888, 2016.MathSciNetzbMATHGoogle Scholar
  64. 64.
    J. Rognes. Galois extensions of structured ring spectra. Stably dualizable groups. Mem. Amer. Math. Soc., 192(898):viii+137, 2008.MathSciNetzbMATHGoogle Scholar
  65. 65.
    R. Rouquier. Dimensions of triangulated categories. J. K-Theory, 1(2):193–256, 2008.MathSciNetzbMATHGoogle Scholar
  66. 66.
    J.-P. Serre. Sur la dimension cohomologique des groupes profinis. Topology, 3:413–420, 1965.MathSciNetzbMATHGoogle Scholar
  67. 67.
    T. Stacks Project Authors. The Stacks Project. Available at
  68. 68.
    A. Vistoli. Grothendieck topologies, fibered categories and descent theory. In Fundamental algebraic geometry, volume 123 of Math. Surveys Monogr., pages 1–104. Amer. Math. Soc., Providence, RI, 2005.Google Scholar
  69. 69.
    E. Yalçin. Set covering and Serre’s theorem on the cohomology algebra of a \(p\)-group. J. Algebra, 245(1):50–67, 2001.MathSciNetzbMATHGoogle Scholar

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© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.University of ChicagoChicagoUSA

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