Nilpotence theorem

Short description: On when an element of the coefficient ring of a ring spectrum is nilpotent

In algebraic topology, the nilpotence theorem gives a condition for an element in the homotopy groups of a ring spectrum to be nilpotent, in terms of the complex cobordism spectrum [math]\displaystyle{ \mathrm{MU} }[/math]. More precisely, it states that for any ring spectrum [math]\displaystyle{ R }[/math], the kernel of the map [math]\displaystyle{ \pi_\ast R \to \mathrm{MU}_\ast(R) }[/math] consists of nilpotent elements. It was conjectured by Douglas Ravenel (1984) and proved by Ethan S. Devinatz, Michael J. Hopkins, and Jeffrey H. Smith (1988).

Nishida's theorem

Goro Nishida (1973) showed that elements of positive degree of the homotopy groups of spheres are nilpotent. This is a special case of the nilpotence theorem.

See also

• Ravenel's conjectures

References

• Devinatz, Ethan S.; Hopkins, Michael J.; Smith, Jeffrey H. (1988), "Nilpotence and stable homotopy theory. I", Annals of Mathematics, Second Series 128 (2): 207–241, doi:10.2307/1971440 
• "The nilpotency of elements of the stable homotopy groups of spheres", Journal of the Mathematical Society of Japan 25 (4): 707–732, 1973, doi:10.2969/jmsj/02540707 .
• Ravenel, Douglas C. (1984), "Localization with respect to certain periodic homology theories", American Journal of Mathematics 106 (2): 351–414, doi:10.2307/2374308, ISSN 0002-9327  Open online version.
• Ravenel, Douglas C. (1992), Nilpotence and periodicity in stable homotopy theory, Annals of Mathematics Studies, 128, Princeton University Press, ISBN 978-0-691-02572-8, https://books.google.com/books?isbn=069102572X 

Further reading

• Connection of X(n) spectra to formal group laws