---
tag: FQ048Z65F9
kind: statement
role: theorem
name: Nishida nilpotence theorem
msc: [55Q45]
uses: [HGZAYPQBXW]
about: [HGZAYPQBXW]
sources:
  - url: https://en.wikipedia.org/wiki/Homotopy_groups_of_spheres
---

## Statement

Let $\pi^s_* = \bigoplus_{n \ge 0} \pi^s_n$ be the graded ring of [[stable homotopy groups of spheres|HGZAYPQBXW]], with product given by composition.
Then every element $\alpha \in \pi^s_n$ with $n > 0$ is nilpotent: $\alpha^m = 0$ for some $m \ge 1$.

## Remarks

Nishida proved the theorem in 1973.
His proof uses extended power constructions on spectra and the operations they induce.
For example, the stable Hopf map $\eta \in \pi^s_1$ satisfies $\eta^4 = 0$, while $\eta^3 \ne 0$.
The hypothesis $n > 0$ is necessary: $\pi^s_0 \cong \Z$ is generated by the identity, and no nonzero integer is nilpotent.
The nilpotence theorem of Devinatz, Hopkins and Smith extends this result from the sphere spectrum to all ring spectra.

## Explanations

### L1 human

Stable maps between spheres can be composed, and composing a map of positive degree with itself raises the degree.
The theorem says that if one keeps composing such a map with itself, the result is eventually null-homotopic.
So no element of positive degree can be iterated forever.

### L2 human

The ring $\pi^s_*$ is graded-commutative, and its positive part consists of torsion elements.
In a general graded-commutative ring, torsion elements need not be nilpotent, so the theorem is a real constraint on the stable stems.
It was the starting point for the chromatic picture of stable homotopy theory, where the nilpotence theorem of Devinatz, Hopkins and Smith characterises nilpotent self-maps through complex cobordism.