K(n)-local stable homotopy theory in nLab
Context
Stable Homotopy theory
Ingredients
Contents
Higher algebra
Algebraic theories
Algebras and modules
Higher algebras
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symmetric monoidal (∞,1)-category of spectra
Model category presentations
Geometry on formal duals of algebras
Theorems
Contents
Idea
The stable homotopy theory of local spectra, local with respect to Morava K-theory K(n)K(n).
Properties
Bilimits
Proposition
For F:X→Sp 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
lim⟶F⟶≃lim⟵F. \underset{\longrightarrow}{\lim} F \stackrel{\simeq}{\longrightarrow} \underset{\longleftarrow}{\lim} F \,.
This is (Hopkins-Lurie 14, theorem 0.0.2).
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 1E→≃L K(n)E. gl_1(E) \longrightarrow L_{K(n)} gl_1E \stackrel{\simeq}{\to} L_{K(n)}E \,.
References
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Mark Hovey, H. Sadofsky, Tate cohomology lowers chromatic Bouseld classes Proceedings of the AMS 124, 1996, 3579-3585.
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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)K(1)-local E ∞E_\infty-Ring spectra (pdf)
Last revised on November 9, 2019 at 19:26:49. See the history of this page for a list of all contributions to it.