nilpotence theorem



Higher linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




A fundamental statement in stable homotopy theory/higher algebra/chromatic homotopy theory.


For any ring spectrum RR the kernel of the canonical morphism

π RMU (R) \pi_\bullet R \longrightarrow MU_\bullet(R)

to the MU-homology of RR consists of nilpotent elements.

This is due to Ethan Devinatz, Mike Hopkins, and Jeff Smith. See Lurie 10, theorem 3

Special cases

The Nishida nilpotence theorem (Nishida 73) is the special case for the sphere spectrum, saying that all positive-degree elements in the stable homotopy groups of spheres are nilpotent.

This is indeed a special case. The MU-homology of the sphere spectrum is the Lazard ring and hence is torsion-free, whereas all positive-degree elements of the stable homotopy ring are torsion by the Serre finiteness theorem and therefore belong to the aforementioned kernel.


Relation with \infty-fields

The nilpotence theorem implies that every ∞-field is an ∞-module over the Morava K-theory spectrum K(n)K(n), for some nn. See at ∞-field.


  • Goro Nishida, (1973), “The nilpotency of elements of the stable homotopy groups of spheres”, Journal of the Mathematical Society of Japan 25 (4): 707–732, doi:10.2969/jmsj/02540707, ISSN 0025-5645, MR 0341485.

  • Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010, Lecture 25 The Nilpotence lemma (pdf)

Last revised on January 17, 2018 at 16:25:03. See the history of this page for a list of all contributions to it.