nLab K(n)-local stable homotopy theory

Idea

The stable homotopy theory of local spectra, local with respect to Morava K-theory $K(n)$.

Properties

Bilimits

Definition

Say that an ∞-groupoid is strictly tame or of finite type (Hopkins-Lurie 14, def. 4.4.1) or maybe better is a truncated homotopy type with finite homotopy groups if it has only finitely many nontrivial homotopy groups each of which is furthermore a finite group.

Proposition

For $F \colon X \to Sp_{K(n)}$ a strictly tame diagram, def. , of $K(n)$-local spectra, then its (∞,1)-limit and (∞,1)-colimit agree in that the canonical comparison map is an equivalence

$$\underset{\longrightarrow}{\lim} F \stackrel{\simeq}{\longrightarrow} \underset{\longleftarrow}{\lim} F \,.$$

This is (Hopkins-Lurie 14, theorem 0.0.2).

Remark

So in particular $K(n)$-local spectra have biproducts, called 0-semiadditivity in (Hopkins-Lurie 14, prop. 4.4.9.

Example

For $X$ pointed connected, hence $X \simeq B G$ the delooping of an ∞-group, a diagram in prop. exhibits an ∞-action of $G$ on some $K(n)$-local spectrum, the (∞,1)-colimit produces the homotopy quotient of the ∞-action and the (∞,1)-limit the homotopy invariants. In this case (Hovey-Sadofsky 96) show that the comparison map exhibits the homotopy invariant as the $K(n)$-localization of the homotopy coinvariants. This in particular means that the comparison map is a $K(n)$-local equivalence, which is the statement of prop. .

Logarithms of twists of generalized cohomology

Let $E$ be an E-∞ ring and write $GL_1(E)$ for its abelian ∞-group of units and $gl_1(E)$ for the corresponding connective spectrum.

Via the Bousfield-Kuhn functor there are natural equivalences between the $K(n)$-localizations of $gl_1(E)$ and $E$ itself.

$$L_{K(n)} gl_1(E) \simeq L_{K(n)} E \,.$$

Composed with the localization map itself, this yields logarithmic cohomology operations

$$gl_1(E) \longrightarrow L_{K(n)} gl_1E \stackrel{\simeq}{\to} L_{K(n)}E \,.$$

Related concepts

• chromatic homotopy theory

• Bousfield-Kuhn functor

• logarithmic cohomology operation

• Bousfield localization of spectra

• KU-local stable homotopy theory

References

• Mark Hovey, H. Sadofsky, Tate cohomology lowers chromatic Bouseld classes Proceedings of the AMS 124, 1996, 3579-3585.

• Michael Hopkins, Jacob Lurie, Ambidexterity in K(n)-Local Stable Homotopy Theory (2014)

Some basics of K(1)-local E-∞ rings are in

• Michael Hopkins, $K(1)$-local $E_\infty$-Ring spectra (pdf)