Nilpotence theorem : Nishida's theorem, Further reading Wikipedia, the free encyclopedia

Nilpotence theorem

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 $\mathrm {MU}$ . 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).

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.

References

^ Lurie, Jacob (April 27, 2010). "The Nilpotence Theorem (Lecture 25)" (PDF). Archived (PDF) from the original on January 30, 2022.

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