nLab
nilpotence theorem
Contents
Context
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

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

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

Statement
For any ring spectrum $R$ the kernel of the canonical morphism

$\pi_\bullet R \longrightarrow MU_\bullet(R)$

to the MU -homology of $R$ consists of nilpotent elements .

This is due to Devinatz-Hopkins-Smith 88 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.

Consequences
Relation with $\infty$ -fields
The nilpotence theorem implies that every ∞-field is an ∞-module over the Morava K-theory spectrum $K(n)$ , for some $n$ . See at ∞-field .

References
Goro Nishida , The nilpotency of elements of the stable homotopy groups of spheres , Journal of the Mathematical Society of Japan 25 (4): 707–732, (1973) (euclid:euclid.jmsj/1240435467 , euclid:jmsj/1240435467 , ISSN 0025-5645, MR 0341485)

Douglas Ravenel , Section 10.1 of: Localization with Respect to Certain Periodic Homology Theories , American Journal of Mathematics Vol. 106, No. 2 (Apr., 1984), pp. 351-414 (doi:10.2307/2374308 , jstor:2374308 )

Ethan Devinatz , Michael Hopkins , Jeffrey Smith , Nilpotence and Stable Homotopy Theory I , Annals of Mathematics Second Series, Vol. 128, No. 2 (Sep., 1988), pp. 207-241 (jstor:1971440 )

Ethan Devinatz , Michael Hopkins , Jeffrey Smith , Nilpotence and Stable Homotopy Theory II , Annals of Mathematics Second Series, Vol. 148, No. 1 (Jul., 1998), pp. 1-49 (jstor:120991 )

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

Last revised on December 30, 2020 at 03:32:46.
See the history of this page for a list of all contributions to it.