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\begin{document}

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\section*{K(n)-local stable homotopy theory}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{stable_homotopy_theory}{}\paragraph*{{Stable Homotopy theory}}\label{stable_homotopy_theory}

[[!include stable homotopy theory - contents]]

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{bilimits}{Bilimits}\dotfill \pageref*{bilimits} \linebreak
\noindent\hyperlink{logarithms_of_twists_of_generalized_cohomology}{Logarithms of twists of generalized cohomology}\dotfill \pageref*{logarithms_of_twists_of_generalized_cohomology} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

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

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{bilimits}{}\subsubsection*{{Bilimits}}\label{bilimits}

\begin{defn}
\label{StrictlyTame}\hypertarget{StrictlyTame}{}
Say that an [[∞-groupoid]] is \emph{[[groupoid cardinality|strictly tame]]} or \emph{of [[finite type]]} (\hyperlink{HopkinsLurie14}{Hopkins-Lurie 14, def. 4.4.1}) or maybe better is a \emph{[[homotopy type with finite homotopy groups|truncated homotopy type with finite homotopy groups]]} if it has only finitely many nontrivial [[homotopy groups]] each of which is furthermore a [[finite group]].

\end{defn}
\begin{prop}
\label{HopkinsLurieTheorem}\hypertarget{HopkinsLurieTheorem}{}
For $F \colon X \to Sp_{K(n)}$ a strictly tame [[diagram]], def. \ref{StrictlyTame}, of $K(n)$-local spectra, then its [[(∞,1)-limit]] and [[(∞,1)-colimit]] agree in that the canonical comparison map is an [[equivalence in an (∞,1)-category|equivalence]]

\begin{displaymath}
\underset{\longrightarrow}{\lim} F 
    \stackrel{\simeq}{\longrightarrow}
  \underset{\longleftarrow}{\lim} F
  \,.
\end{displaymath}
\end{prop}
This is (\hyperlink{HopkinsLurie14}{Hopkins-Lurie 14, theorem 0.0.2}).

\begin{remark}
\label{}\hypertarget{}{}
So in particular $K(n)$-local spectra have \emph{[[biproducts]]}, called \emph{[[0-semiadditivity]]} in (\hyperlink{HopkinsLurie14}{Hopkins-Lurie 14, prop. 4.4.9}.

\end{remark}
\begin{example}
\label{}\hypertarget{}{}
For $X$ pointed [[connected]], hence $X \simeq B G$ the [[delooping]] of an [[∞-group]], a [[diagram]] in prop. \ref{HopkinsLurieTheorem} 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 (\hyperlink{HoveySadofsky96}{Hovey-Sadofsky 96}) show that the comparison map exhibits the homotopy invariant as the $K(n)$-[[localization of a spectrum|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. \ref{HopkinsLurieTheorem}.

\end{example}
\hypertarget{logarithms_of_twists_of_generalized_cohomology}{}\subsubsection*{{Logarithms of twists of generalized cohomology}}\label{logarithms_of_twists_of_generalized_cohomology}

Let $E$ be an [[E-∞ ring]] and write $GL_1(E)$ for its [[abelian ∞-group|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.

\begin{displaymath}
L_{K(n)} gl_1(E) \simeq L_{K(n)} E
  \,.
\end{displaymath}
Composed with the [[Bousfield localization of spectra|localization map]] itself, this yields [[logarithmic cohomology operations]]

\begin{displaymath}
gl_1(E) \longrightarrow L_{K(n)} gl_1E \stackrel{\simeq}{\to} L_{K(n)}E
  \,.
\end{displaymath}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[chromatic homotopy theory]]


\item [[Bousfield-Kuhn functor]]


\item [[logarithmic cohomology operation]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Mark Hovey]], H. Sadofsky, \emph{Tate cohomology lowers chromatic Bouseld classes} Proceedings of the AMS 124, 1996, 3579-3585.


\item [[Michael Hopkins]], [[Jacob Lurie]], \emph{[[Ambidexterity in K(n)-Local Stable Homotopy Theory]]} (2014)



\end{itemize}
Some basics of [[Morava K-theory|K(1)]]-local [[E-∞ rings]] are in

\begin{itemize}%
\item [[Michael Hopkins]], \emph{$K(1)$-local $E_\infty$-Ring spectra} (\href{http://www.math.rochester.edu/people/faculty/doug/otherpapers/knlocal.pdf}{pdf})

\end{itemize}
[[!redirects K(n)-local spectrum]] [[!redirects K(n)-local spectra]]

[[!redirects K(n)-localization]] [[!redirects K(n)-localizations]]



\end{document}