Math 2.0

Algebraic topology Homotopy groups Theorem

Nilpotence theorem

Builds on Ring spectrumComplex cobordismNishida nilpotence theorem

Statement

Let RR be a homotopy associative ring spectrum, and let MUMU be the spectrum of complex cobordism. Let h ⁣:π∗R→MU∗Rh \colon \pi_* R \to MU_* R be the Hurewicz homomorphism, which sends α\alpha to the image of α\alpha under the unit map R→MU∧RR \to MU \wedge R. Then every element of the kernel of hh is nilpotent.

Remarks

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

Going deeper

For a student

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 KK-theory type. These results proved most of the Ravenel conjectures.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Complex cobordism, concept, no evidence levelComplex cobordismconceptRing spectrum, concept, no evidence levelRing spectrumconceptStable homotopy groups of spheres, concept, no evidence levelStable homotopy group…conceptNishida nilpotence theorem, statement, no evidence levelNishida nilpotence th…statementNilpotence theorem, statement, no evidence levelNilpotence theoremstatement
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  1. Complex cobordism concept
  2. Ring spectrum concept
  3. Stable homotopy groups of spheres concept
  4. Nishida nilpotence theorem statement
  5. Nilpotence theorem statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Nilpotence_theorem
About this page
Tag9N027RT5R6
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Kindstatement, theorem
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MSC55Q45, 55P42, 55N22
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AboutRing spectrum, Complex cobordism
UsesRing spectrum, Complex cobordism, Nishida nilpotence theorem
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Changed2 November 2026

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