symmetric monoidal (∞,1)-category of spectra
The stable homotopy theory of local spectra, local with respect to Morava K-theory .
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.
For a strictly tame diagram, def. , of -local spectra, then its (∞,1)-limit and (∞,1)-colimit agree in that the canonical comparison map is an equivalence
This is (Hopkins-Lurie 14, theorem 0.0.2).
So in particular -local spectra have biproducts, called 0-semiadditivity in (Hopkins-Lurie 14, prop. 4.4.9.
For pointed connected, hence the delooping of an ∞-group, a diagram in prop. exhibits an ∞-action of on some -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 -localization of the homotopy coinvariants. This in particular means that the comparison map is a -local equivalence, which is the statement of prop. .
Let be an E-∞ ring and write for its abelian ∞-group of units and for the corresponding connective spectrum.
Via the Bousfield-Kuhn functor there are natural equivalences between the -localizations of and itself.
Composed with the localization map itself, this yields logarithmic cohomology operations
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
Last revised on November 9, 2019 at 19:26:49. See the history of this page for a list of all contributions to it.