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K(n)-local stable homotopy theory

Contents

Context

Stable Homotopy theory

Higher algebra

Contents

Idea

The stable homotopy theory of local spectra, local with respect to Morava K-theory K(n)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:XSp K(n)F \colon X \to Sp_{K(n)} a strictly tame diagram, def. , of K(n)K(n)-local spectra, then its (∞,1)-limit and (∞,1)-colimit agree in that the canonical comparison map is an equivalence

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

Example

For XX pointed connected, hence XBGX \simeq B G the delooping of an ∞-group, a diagram in prop. exhibits an ∞-action of GG on some K(n)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)K(n)-localization of the homotopy coinvariants. This in particular means that the comparison map is a K(n)K(n)-local equivalence, which is the statement of prop. .

Logarithms of twists of generalized cohomology

Let EE be an E-∞ ring and write GL 1(E)GL_1(E) for its abelian ∞-group of units and gl 1(E)gl_1(E) for the corresponding connective spectrum.

Via the Bousfield-Kuhn functor there are natural equivalences between the K(n)K(n)-localizations of gl 1(E)gl_1(E) and EE itself.

L K(n)gl 1(E)L K(n)E. 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)L K(n)gl 1EL K(n)E. gl_1(E) \longrightarrow L_{K(n)} gl_1E \stackrel{\simeq}{\to} L_{K(n)}E \,.

References

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

Last revised on August 15, 2014 at 08:53:21. See the history of this page for a list of all contributions to it.