nLab infinity-field

Idea

The notion of $A_\infty$-field is to the notion of A-∞ algebra as that of field is to associative algebra/ring.

Definition

Definition

An A-∞ ring or in fact just an H-space $A$ is a field if $\pi_\bullet A$ is a graded field.

For instance (Lurie, lecture 24, def. 3).

Properties

For $E$ an $\infty$-field, def. , then it carries the structure of an ∞-module over the $n$th Morava K-theory spectrum $K(n)$, for some $n$.

(Lurie 10, lect 25, cor 10)

This follows with the nilpotence theorem.

Examples

• For $k$ an ordinary field, the Eilenberg-Mac Lane spectrum of $k$, $H k$, is an $A_\infty$-field.
• The Morava K-theory A-∞ rings $K(n)$ are the basic $A_\infty$-fields. See at Morava K-theory – As infinity-Fields, where $K(0) \simeq H \mathbb{Q}$ and we define $K(\infty)$ as $H \mathbb{F}_p$.

Related concepts

• chromatic homotopy theory

References

Definition 3 in

• Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010, Lecture 24 Uniqueness of Morava K-theory (pdf)

• Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010,

  Lecture 25 The Nilpotence lemma (pdf)