Tunisian Journal of Mathematics

Nilpotence theorems via homological residue fields

Paul Balmer

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/tunis.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We prove nilpotence theorems in tensor-triangulated categories using suitable Gabriel quotients of the module category, and discuss examples.

Article information

Tunisian J. Math., Volume 2, Number 2 (2020), 359-378.

Received: 1 October 2018
Revised: 22 January 2019
Accepted: 14 March 2019
First available in Project Euclid: 13 August 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 18E30: Derived categories, triangulated categories
Secondary: 20J05: Homological methods in group theory 55U35: Abstract and axiomatic homotopy theory

Homological residue field tensor-triangular geometry module category nilpotence


Balmer, Paul. Nilpotence theorems via homological residue fields. Tunisian J. Math. 2 (2020), no. 2, 359--378. doi:10.2140/tunis.2020.2.359. https://projecteuclid.org/euclid.tunis/1565661721

Export citation


  • P. Balmer, “The spectrum of prime ideals in tensor triangulated categories”, J. Reine Angew. Math. 588 (2005), 149–168.
  • P. Balmer, “Spectra, spectra, spectra-tensor triangular spectra versus Zariski spectra of endomorphism rings”, Algebr. Geom. Topol. 10:3 (2010), 1521–1563.
  • P. Balmer and B. Sanders, “The spectrum of the equivariant stable homotopy category of a finite group”, Invent. Math. 208:1 (2017), 283–326.
  • P. Balmer, H. Krause, and G. Stevenson, “The frame of smashing tensor-ideals”, Math. Proc. Cambridge Philos. Soc. (online publication October 2018), 1–21.
  • P. Balmer, H. Krause, and G. Stevenson, “Tensor-triangular fields: ruminations”, Selecta Math. $($N.S.$)$ 25:1 (2019), 25:13.
  • T. Barthel, J. P. C. Greenlees, and M. Hausmann, “On the Balmer spectrum for compact Lie groups”, preprint, 2018.
  • D. J. Benson, J. F. Carlson, and J. Rickard, “Thick subcategories of the stable module category”, Fund. Math. 153:1 (1997), 59–80.
  • D. Benson, S. B. Iyengar, H. Krause, and J. Pevtsova, “Stratification for module categories of finite group schemes”, J. Amer. Math. Soc. 31:1 (2018), 265–302.
  • E. S. Devinatz, M. J. Hopkins, and J. H. Smith, “Nilpotence and stable homotopy theory, I”, Ann. of Math. $(2)$ 128:2 (1988), 207–241.
  • E. M. Friedlander and J. Pevtsova, “$\Pi$-supports for modules for finite group schemes”, Duke Math. J. 139:2 (2007), 317–368.
  • P. Gabriel, “Des catégories abéliennes”, Bull. Soc. Math. France 90 (1962), 323–448.
  • M. J. Hopkins, “Global methods in homotopy theory”, pp. 73–96 in Homotopy theory (Durham, NC, 1985), edited by E. Rees and J. D. S. Jones, London Math. Soc. Lecture Note Ser. 117, Cambridge Univ. Press, 1987.
  • M. J. Hopkins and J. H. Smith, “Nilpotence and stable homotopy theory, II”, Ann. of Math. $(2)$ 148:1 (1998), 1–49.
  • H. Krause, “Smashing subcategories and the telescope conjecture–-an algebraic approach”, Invent. Math. 139:1 (2000), 99–133.
  • H. Krause, “Cohomological quotients and smashing localizations”, Amer. J. Math. 127:6 (2005), 1191–1246.
  • A. Mathew, “Residue fields for a class of rational $\bold{E}_\infty$-rings and applications”, J. Pure Appl. Algebra 221:3 (2017), 707–748.
  • A. Neeman, “The chromatic tower for $D(R)$”, Topology 31:3 (1992), 519–532.
  • A. Neeman, “Oddball Bousfield classes”, Topology 39:5 (2000), 931–935.
  • A. Neeman, Triangulated categories, Annals of Mathematics Studies 148, Princeton University Press, 2001.
  • D. C. Ravenel, Nilpotence and periodicity in stable homotopy theory, Annals of Mathematics Studies 128, Princeton University Press, 1992.
  • R. W. Thomason, “The classification of triangulated subcategories”, Compositio Math. 105:1 (1997), 1–27.