Nishida nilpotence theorem

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.

Going deeper

For a student

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.