nLab nilpotence theorem

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.

Related concepts

• Serre finiteness theorem

• thick subcategory theorem

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)

• 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)