nLab infinity-field (changes)

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Context

Higher linear algebra

Idea
Definition
Properties
Examples
Related concepts
References

Idea

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

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. 1, 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

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 $H \mathbb{F}_p$ .

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)

Last revised on August 20, 2014 at 23:57:09. See the history of this page for a list of all contributions to it.

nLab infinity-field (changes)

Context

Higher linear algebra

Contents

Idea

Definition

Definition

Properties

Examples

Related concepts

References