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 $\mathrm{MU}$ . More precisely, it states that for any ring spectrum $R$ , the kernel of the map $\pi_\ast R \to \mathrm{MU}_\ast(R)$ consists of nilpotent elements.{{Cite web |last=Lurie |first=Jacob |date=April 27, 2010 |title=The Nilpotence Theorem (Lecture 25) |url=https://www.math.ias.edu/~lurie/252xnotes/Lecture25.pdf |url-status=live |archive-url=https://web.archive.org/web/20220130234323/https://www.math.ias.edu/~lurie/252xnotes/Lecture25.pdf |archive-date=January 30, 2022 |access-date=}} It was conjectured by {{harvs|txt|last=Ravenel | first=Douglas |authorlink=Douglas Ravenel| year=1984}} and proved by {{harvs|txt| last1=Devinatz | first1=Ethan S. | last2=Hopkins | first2=Michael J. | authorlink2=Michael J. Hopkins | last3=Smith | first3=Jeffrey H. |year=1988}}.

Nishida's theorem

{{harvs|txt|last=Nishida|first= Goro|authorlink=Goro Nishida|year=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

{{Citation | last1=Devinatz | first1=Ethan S. | last2=Hopkins | first2=Michael J. | authorlink2=Michael J. Hopkins | last3=Smith | first3=Jeffrey H. | title=Nilpotence and stable homotopy theory. I | doi=10.2307/1971440 | mr=960945 | year=1988 | journal=Annals of Mathematics |series=Second Series | volume=128 | issue=2 | pages=207–241| jstor=1971440 }}
{{citation

|doi= 10.2969/jmsj/02540707

|last= Nishida | first= Goro | authorlink = Goro Nishida

|title= The nilpotency of elements of the stable homotopy groups of spheres

|journal= Journal of the Mathematical Society of Japan

|volume= 25

|issue= 4

|year= 1973

|pages= 707–732

|mr= 0341485

|doi-access= free

|hdl= 2433/220059

|hdl-access= free

}}.

{{Citation | last=Ravenel | first=Douglas C. |authorlink=Douglas Ravenel| title=Localization with respect to certain periodic homology theories | doi=10.2307/2374308 | mr=737778 | year=1984 | journal=American Journal of Mathematics | issn=0002-9327 | volume=106 | issue=2 | pages=351–414| jstor=2374308 }} [https://web.archive.org/web/20120308131453/http://www.math.rochester.edu/u/faculty/doug/mypapers/loc.pdf Open online version.]
{{Citation | last=Ravenel | first=Douglas C. | title=Nilpotence and periodicity in stable homotopy theory | url=https://books.google.com/books?isbn=069102572X | publisher=Princeton University Press | series=Annals of Mathematics Studies | isbn=978-0-691-02572-8 | mr=1192553 | year=1992 | volume=128}}

nilpotence theorem

Nishida's theorem

See also

References

Further reading