Contents

# Contents

## 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 Ethan Devinatz, Mike Hopkins, and Jeff Smith. 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.

## References

• Goro Nishida, (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, ISSN 0025-5645, MR 0341485.

• Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010, Lecture 25 The Nilpotence lemma (pdf)

Last revised on January 17, 2018 at 16:25:03. See the history of this page for a list of all contributions to it.