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 ${\displaystyle \mathrm {MU} }$. More precisely, it states that for any ring spectrum ${\textstyle R}$, the kernel of the map ${\textstyle \pi _{\ast }R\to \mathrm {MU} _{\ast }(R)}$ consists of nilpotent elements. ^[1] It was conjectured by Douglas Ravenel ( 1984) and proved by Ethan S. Devinatz, Michael J. Hopkins, and Jeffrey H. Smith ( 1988).

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.

• Ravenel's conjectures

• 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, JSTOR  1971440, MR  0960945
• Nishida, Goro (1973), "The nilpotency of elements of the stable homotopy groups of spheres", Journal of the Mathematical Society of Japan, 25 (4): 707–732, doi: 10.2969/jmsj/02540707, hdl: 2433/220059, MR  0341485.
• 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, JSTOR  2374308, MR  0737778 Open online version.
• Ravenel, Douglas C. (1992), Nilpotence and periodicity in stable homotopy theory, Annals of Mathematics Studies, vol. 128, Princeton University Press, ISBN  978-0-691-02572-8, MR  1192553

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

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