---
tag: 9N027RT5R6
kind: statement
role: theorem
name: Nilpotence theorem
msc: [55Q45, 55P42, 55N22]
uses: [BAW0T68GWK, 5CZHFNY9F3, FQ048Z65F9]
about: [BAW0T68GWK, 5CZHFNY9F3]
sources:
  - url: https://en.wikipedia.org/wiki/Nilpotence_theorem
same_as:
  wikidata: Q17099400
  wikipedia: Nilpotence theorem
---

## Statement

Let $R$ be a homotopy associative [[ring spectrum|BAW0T68GWK]], and let $MU$ be the spectrum of [[complex cobordism|5CZHFNY9F3]].
Let $h \colon \pi_* R \to MU_* R$ be the Hurewicz homomorphism, which sends $\alpha$ to the image of $\alpha$ under the unit map $R \to MU \wedge R$.
Then every element of the kernel of $h$ is nilpotent.

## Remarks

The theorem was proved by Devinatz, Hopkins and Smith in 1988.
The proof filters $MU$ by a sequence of Thom spectra $X(n)$ built from $\Omega SU(n)$ and shows inductively that detection of nilpotence passes from $X(n+1)$ to $X(n)$.
For $R$ the sphere spectrum, $MU_* = \pi_* MU$ is torsion-free while every element of positive degree in $\pi_* S$ is torsion.
So the Hurewicz image of such an element is zero, and the theorem gives back the [[Nishida nilpotence theorem|FQ048Z65F9]].
Equivalent forms state that a self-map $\Sigma^d X \to X$ of a finite spectrum is nilpotent when its $MU$-homology is nilpotent.

## Explanations

### L1 human

Complex cobordism $MU$ is a homology theory that sees a lot of stable homotopy theory.
The theorem says that whatever $MU$ cannot see in a ring spectrum must be nilpotent.
So $MU$ detects all the elements that can be iterated forever.

### L2 human

The nilpotence theorem is the foundation of chromatic homotopy theory.
Together with the work of Hopkins and Smith it yields the thick subcategory theorem and the periodicity theorem, which classify finite spectra by their Morava $K$-theory type.
These results proved most of the Ravenel conjectures.