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\begin{document}

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\section*{Bousfield localization of spectra}

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

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{localization_at_moore_spectra_of_abelian_groups}{Localization at Moore spectra of abelian groups}\dotfill \pageref*{localization_at_moore_spectra_of_abelian_groups} \linebreak
\noindent\hyperlink{relation_to_nilpotent_completion}{Relation to nilpotent completion}\dotfill \pageref*{relation_to_nilpotent_completion} \linebreak
\noindent\hyperlink{fracture_theorem}{Fracture theorem}\dotfill \pageref*{fracture_theorem} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{rationalization}{Rationalization}\dotfill \pageref*{rationalization} \linebreak
\noindent\hyperlink{localization}{$p$-Localization}\dotfill \pageref*{localization} \linebreak
\noindent\hyperlink{completion}{$p$-Completion}\dotfill \pageref*{completion} \linebreak
\noindent\hyperlink{telescopic_localization}{Telescopic localization}\dotfill \pageref*{telescopic_localization} \linebreak
\noindent\hyperlink{chromatic_localization}{Chromatic localization}\dotfill \pageref*{chromatic_localization} \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}

\emph{Bousfield localization of spectra} refers generally to [[localization of an (∞,1)-category|localizations]] of the [[stable (∞,1)-category of spectra]] (hence [[Bousfield localization of model categories|Bousfield localization]] of [[model categories of spectra]]) at the collection of [[morphisms]] which become [[equivalences]] under [[smash product of spectra|smash product]] with a given [[spectrum]] $E$. Since any such $E$ represents a [[generalized homology theory]], this may also be thought of \emph{$E$-[[homology localization]]}.

More specifically, if the [[stable (∞,1)-category of spectra]] is [[presentable (infinity,1)-category|presented]] by a ([[stable model category|stable]]) [[model category]], then the [[localization of an (∞,1)-category|∞-categorical localization]] can be presented by the operation of \emph{[[Bousfield localization of model categories]]}. The original article (\hyperlink{Bousfield79}{Bousfield 79}) essentially considered localization at the level of [[homotopy categories]].

Specifically, for $E \in Spec$ a [[spectrum]], the \emph{Bousfield localization at $E$} of another [[spectrum]] $X$ is the [[universal property|universal map]]

\begin{displaymath}
X \longrightarrow L_E X
\end{displaymath}
to the \emph{$E$-[[local spectrum]]} $L_E X$, with the property that for $Y$ any $E$-acyclic spectrum in that $Y \wedge E \simeq 0$, every morphism $Y \longrightarrow L_E X$ is null-homotopic (a [[zero morphism]] in the [[stable (∞,1)-category of spectra]]). (see for instance \hyperlink{Lurie10}{Lurie 10, Example 4})

For $E = M \mathbb{Z}_p$ the [[Moore spectrum]] of the [[cyclic group]] $\mathbb{Z}_p \coloneqq \mathbb{Z}/p\mathbb{Z}$ for some [[prime number]] $p$, this $E$-localization is \emph{[[p-completion]]}. (see for instance \hyperlink{Lurie10}{Lurie 10, Examples 7 and 8})

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

We write $Ho(Spectra)$ for the [[stable homotopy category]]. For $X,Y \in Ho(Spectra)$ two [[spectra]] we write $[X,Y] \in$ [[Ab]] for its [[hom-object|hom-]][[abelian groups]], and $[X,Y]_\bullet \coloneqq [\Sigma^\bullet X,Y]$ for the corresponding [[graded abelian group]] (\href{Introduction+to+Stable+homotopy+theory+--+1-1#GradedAbelianGroupStructureOnHomsInTheHomotopyCategory}{def.}).

\begin{defn}
\label{EAcyclicAndLocalSpectra}\hypertarget{EAcyclicAndLocalSpectra}{}
Let $E \in Ho(Spectra)$ be a [[spectrum]]. Say that

\begin{enumerate}%
\item a spectrum $X$ is \textbf{$E$-acyclic} if the [[smash product of spectra|smash product]] with $E$ is [[zero object|zero]], $E \wedge X \simeq 0$;


\item a morphism $f \colon X \to Y$ of spectra is an \textbf{$E$-equivalence} if $E \wedge f \;\colon\; E \wedge X \to E \wedge Y$ is an [[isomorphism]] in $Ho(Spectra)$, hence if $E_\bullet(f)$ is an isomorphism in $E$-[[generalized homology]];


\item a spectrum $X$ is \textbf{$E$-local} if the following equivalent conditions hold

\begin{enumerate}%
\item for every $E$-equivalence $f$ then $[f,X]_\bullet$ is an isomorphism;


\item every [[morphism]] $Y \longrightarrow X$ out of an $E$-acyclic spectrum $Y$ is [[zero morphism|zero]] in $Ho(Spectra)$;



\end{enumerate}


\end{enumerate}
\end{defn}
(\hyperlink{Bousfield79}{Bousfield 79, \S{}1}) see also for instance (\hyperlink{Lurie10}{Lurie, Lecture 20, example 4})

\begin{lemma}
\label{TwoConditionsOnELocalSpectrumAreEquivalent}\hypertarget{TwoConditionsOnELocalSpectrumAreEquivalent}{}
The two conditions in the last item of def. \ref{EAcyclicAndLocalSpectra} are indeed equivalent.

\end{lemma}
\begin{proof}
Notice that $A \in Ho(Spectra)$ being $E$-acyclic means equivalently that the unique morphism $0 \longrightarrow A$ is an $E$-equivalence.

Hence one direction of the claim is trivial. For the other direction we need to show that for $[-,X]_\bullet$ to give an isomorphism on all $E$-equivalences $f$, it is sufficient that it gives an isomorphism on all $E$-equivalences of the form $0 \to A$.

Given a morphism $f \colon A \to B$, write $B \longrightarrow B/A$ for its [[homotopy cofiber]]. Then since $Ho(Spectra)$ is a [[triangulated category]] (\href{Introduction+to+Stable+homotopy+theory+--+1-1#StableHomotopyCategoryIsTriangulated}{prop.}) the defining axioms of triangulated categories (\href{Introduction+to+Stable+homotopy+theory+--+1-1#CategoryWithCofiberSequences}{def.}, \href{Introduction+to+Stable+homotopy+theory+--+1-1#TrianglesMayBeShiftedToTheLeft}{lemma}) give that there is a [[commuting diagram]] of the form

\begin{displaymath}
\itexarray{
    0
      &\longrightarrow&
    A 
      &\overset{id}{\longrightarrow}&
    A
      &\overset{}{\longrightarrow}&
    0
      &\overset{}{\longrightarrow}&
    \Sigma A
    \\
    \downarrow
      &&
    \downarrow^{\mathrlap{id}}
      &&
    \downarrow^{\mathrlap{f}}
      &&
    \downarrow
      &&
    \downarrow^{\mathrlap{id}}
    \\
    \Sigma^{-1} B/A
      &\longrightarrow&
    A 
      &\underset{f}{\longrightarrow}&
    B
      &\longrightarrow&
    B/A
      &\longrightarrow&
    \Sigma A
  }
  \,,
\end{displaymath}
where both the top as well as the bottom are [[homotopy cofiber sequences]]. Hence applying $[-,X]_\bullet$ to this diagram in $Ho(Spectra)$ yields a diagram of [[graded abelian groups]] of the form

\begin{displaymath}
\itexarray{
    0 
      &\longleftarrow&
    [A,X]_\bullet
      &\longleftarrow&
    [A,X]_\bullet
      &\longleftarrow&
    0
      &\longleftarrow&
    [A,X]_{\bullet+1}
    \\
    \uparrow
      &&
    \uparrow^{\mathrlap{id}}
      &&
    \uparrow^{\mathrlap{[f,X]_\bullet}}
      &&
    \uparrow
      &&
    \uparrow^{\mathrlap{id}}
    \\
    [B/A,X]_{\bullet+1}
      &\longleftarrow&
    [A,X]_\bullet
      &\longleftarrow&
    [B,X]_\bullet
      &\longleftarrow&
    [B/A,X]_\bullet
      &\longleftarrow&
    [A,X]_{\bullet+1}
  }
  \,,
\end{displaymath}
where now both horizontal sequences are [[long exact sequences]] (\href{Introduction+to+Stable+homotopy+theory+--+1-1#LongFiberSequencesOfMapsOfSpectra}{prop.}).

Hence if $[B/A,X]_\bullet \longrightarrow 0$ is an isomorphism, then all four outer vertical morphisms in this diagram are isomorphisms, and then the [[five-lemma]] implies that also $[f,X]_\bullet$ is an isomorphism.

Hence it is now sufficient to observe that with $f \colon A \to B$ an $E$-equivalence, then its homotopy cofiber $B/A$ is $E$-acyclic.

To see this, notice that by the [[tensor triangulated category|tensor triangulated]] structure on $Ho(Spectra)$ (\href{Introduction+to+Stable+homotopy+theory+--+1-2#TensorTriangulatedStructureOnStableHomotopyCategory}{prop.}) the [[smash product of spectra|smash product]] with $E$ preserves homotopy cofiber sequences, so that there is a homotopy cofiber sequence

\begin{displaymath}
E \wedge A
    \overset{E \wedge f}{\longrightarrow}
  E \wedge B
    \longrightarrow
  E \wedge (B/A)
    \longrightarrow
  E \wedge \Sigma A
  \,.
\end{displaymath}
But if the first morphism here is an isomorphism, then the axioms of a [[triangulated category]] (\href{Introduction+to+Stable+homotopy+theory+--+1-1#CategoryWithCofiberSequences}{def.}) imply that $E \wedge B / A \simeq 0$. In detail: by the axioms we may form the morphism of homotopy cofiber sequences

\begin{displaymath}
\itexarray{
    E \wedge A
      &\overset{E \wedge f}{\longrightarrow}&
    E \wedge B
      &\longrightarrow&
    E \wedge B/A
      &\longrightarrow&
    E \wedge \Sigma A
    \\
    \downarrow^{\mathrlap{id}}
      &&
    \downarrow^{\mathrlap{(E\wedge f)^{-1}}}
      &&
    \downarrow
      &&
    \downarrow^{\mathrlap{id}}
    \\
    E \wedge A
      &\underset{id}{\longrightarrow}&
    E \wedge A
      &\longrightarrow&
    0
      &\longrightarrow&
    E \wedge \Sigma A
  }
  \,.
\end{displaymath}
Then since two of the three vertical morphisms on the left are isomorphisms, so is the third (\href{Introduction+to+Stable+homotopy+theory+--+1-1#TwoOutOfThreeForMorphismsOfDistinguishedTriangles}{lemma}).

\end{proof}
\begin{defn}
\label{ELocalizationOfSpectra}\hypertarget{ELocalizationOfSpectra}{}
Given $E,X \in Ho(Spectra)$, then an \textbf{$E$-localization} of $X$ is

\begin{enumerate}%
\item an $E$-local spectrum $L_E X$


\item an $E$-equivalence $X \longrightarrow L_E X$.



\end{enumerate}
according to def. \ref{EAcyclicAndLocalSpectra}.

\end{defn}
We discuss now that $E$-Localizations always exist. The key to this is the following lemma \ref{KappaCellSpectrumWitnessingELocalization}, which asserts that a spectrum being $E$-local is equivalent to it being $A$-null, for some ``small'' spectrum $A$:

\begin{lemma}
\label{KappaCellSpectrumWitnessingELocalization}\hypertarget{KappaCellSpectrumWitnessingELocalization}{}
For every [[spectrum]] $E$ there exists a spectrum $A$ such that any spectrum $X$ is $E$-local (def. \ref{EAcyclicAndLocalSpectra}) precisely if it is $A$-null, i.e.

\begin{displaymath}
X \;is\; E\text{-local}
    \;\;\;\;
  \Leftrightarrow
    \;\;\;\;
  [A,X]_\ast = 0
\end{displaymath}
and such that

\begin{enumerate}%
\item $A$ is $E$-acyclic (def. \ref{EAcyclicAndLocalSpectra});


\item there exists an infinite [[cardinal number]] $\kappa$ such that $A$ is a $\kappa$-[[CW spectrum]] (hence a [[CW spectrum]] (\href{Introduction+to+Stable+homotopy+theory+--+1-1#CWSpectrum}{def.}) with at most $\kappa$ many cells);


\item the class of $E$-acyclic spectra (def. \ref{EAcyclicAndLocalSpectra}) is the class generated by $A$ under

\begin{enumerate}%
\item [[wedge sum]]


\item the relation that if in a [[homotopy cofiber sequence]] $X_1 \to X_2 \to X_3$ two of the spectra are in the class, then so is the third.



\end{enumerate}


\end{enumerate}
\end{lemma}
(\hyperlink{Bousfield79}{Bousfield 79, lemma 1.13 with lemma 1.14}) review includes (\href{Bauer11}{Bauer 11, p.2,3}, \hyperlink{VanKoughnett13}{VanKoughnett 13, p. 8})

\begin{prop}
\label{ELocalizationCofiber}\hypertarget{ELocalizationCofiber}{}
For $E \in Ho(Spectra)$ any [[spectrum]], every spectrum $X$ sits in a [[homotopy cofiber sequence]] of the form

\begin{displaymath}
G_E(X) 
    \longrightarrow 
  X 
    \overset{\eta_X}{\longrightarrow}
  L_E(X)
  \,,
\end{displaymath}
and [[natural transformation|natural]] in $X$, such that

\begin{enumerate}%
\item $G_E(X)$ is $E$-acyclic,


\item $L_E(X)$ is $E$-local,



\end{enumerate}
according to def. \ref{EAcyclicAndLocalSpectra}.

\end{prop}
(\hyperlink{Bousfield79}{Bousfield 79, theorem 1.1}) see also for instance (\hyperlink{Lurie10}{Lurie, Lecture 20, example 4})

\begin{proof}
Consider the $\kappa$-[[CW-spectrum]] spectrum $A$ whose existence is asserted by lemma \ref{KappaCellSpectrumWitnessingELocalization}. Let

\begin{displaymath}
I_A
  \coloneqq
  \{A \to Cone(A)\}
\end{displaymath}
denote the set containing as its single element the canonical morphism (of [[sequential spectra]]) from $A$ into the standard [[cone]] of $A$, i.e. the cofiber

\begin{displaymath}
Cone(A) 
    \coloneqq 
  cofib( A \to A \wedge I_+ )
    \simeq
  A \wedge I
\end{displaymath}
of the inclusion of $A$ into its standard [[cylinder spectrum]] (\href{Introduction+to+Stable+homotopy+theory+--+1-1#StandardCylinderSpectrumSequential}{def.}).

Since the standard cylinder spectrum on a CW-spectrum is a [[good cylinder object]] (\href{Introduction+to+Stable+homotopy+theory+--+1-1#CylinderSpectrumOverCWSpectrumIsGood}{prop.}) this means (\href{Introduction+to+Stable+homotopy+theory+--+P#HomsOutOfCofibrantIntoFibrantComputeHomotopyCategory}{lemma}) that for $X$ any fibrant sequential spectrum, and for $A \longrightarrow X$ any morphism, then an extension along the cone inclusion

\begin{displaymath}
\itexarray{
    A &\longrightarrow& X
    \\
    \downarrow & \nearrow
    \\
    Cone(A)
  }
\end{displaymath}
equivalently exhibits a null-homotopy of the top morphism. Hence the $(A \to Cone(A))$-[[injective objects]] in $Ho(Spectra)$ are precisely those spectra $X$ for which $[A,X]_\bullet \simeq 0$.

Moreover, due to the degreewise nature of the smash tensoring $Cone(A) = A \wedge I$ (\href{Introduction+to+Stable+homotopy+theory+--+1-1#TensoringAndPoweringOfSequentialSpectra}{def}), the inclusion morphism $A \to Cone(A)$ is degreewise the inclusion of a [[CW-complex]] into its standard cone, which is a [[relative cell complex]] inclusion (\href{Introduction+to+Stable+homotopy+theory+--+P#TopologicalCylinderOnCWComplexIsGoodCylinderObject}{prop.}).

By \href{Introduction+to+Stable+homotopy+theory+--+1-1#kappaCellSpectrumIsKappaSmall}{this lemma} the $\kappa$-[[cell spectrum]] $A$ is \emph{$\kappa$-small object} (\href{Introduction+to+Stable+homotopy+theory+--+P#ClassOfMorphismsWithSmallDomains}{def.}) with respect to morphisms of spectra which are degreewise [[relative cell complex]] inclusion [[small object argument]] .

Hence the [[small object argument]] applies (\href{Introduction+to+Stable+homotopy+theory+--+P#SmallObjectArgument}{prop.}) and gives for every $X$ a factorization of the terminal morphism $X \to \ast$ as an $I_A$-[[relative cell complex]] (\href{Introduction+to+Stable+homotopy+theory+--+P#TopologicalCCellComplex}{def.}) followed by an $I_A$-[[injective morphism]] (\href{Introduction+to+Stable+homotopy+theory+--+P#RightLiftingProperty}{def.})

\begin{displaymath}
X
    \overset{I_A Cell}{\longrightarrow} 
  L_E X
    \overset{I_A Inj}{\longrightarrow}
  \ast
  \,.
\end{displaymath}
By the above, this means that $[A, L_E X] = 0$, hence by lemma \ref{KappaCellSpectrumWitnessingELocalization} that $L_E X$ is $E$-local.

It remains to see that the [[homotopy fiber]] of $X \to L_E X$ is $E$-acyclic: By the [[tensor triangulated category|tensor triangulated]] structure on $Ho(Spectra)$ (\href{Introduction+to+Stable+homotopy+theory+--+1-2#TensorTriangulatedStructureOnStableHomotopyCategory}{prop.}) it is sufficient to show that the [[homotopy cofiber]] is $E$-acyclic (since it differs from the homotopy fiber only by suspension). By the [[pasting law]], the homotopy cofiber of a [[transfinite composition]] is the transfinite composition of a sequence of homotopy pushouts. By lemma \ref{KappaCellSpectrumWitnessingELocalization} and applying the pasting law again, all these homotopy pushouts produce $E$-acyclic objects. Hence we conclude by observing that the the transfinite composition of the morphisms between these $E$-acyclic objects is $E$-acyclic. Since by construction all these morphisms are relative cell complex inclusions, this follows again with the compactness of the $n$-spheres (\href{Introduction+to+Stable+homotopy+theory+--+P#CompactSubsetsAreSmallInCellComplexes}{lemma}).

\end{proof}
\begin{lemma}
\label{}\hypertarget{}{}
The morphism $X \to L_E (X)$ in prop. \ref{ELocalizationCofiber} exhibits an $E$-localization of $X$ according to def. \ref{ELocalizationOfSpectra}

\end{lemma}
\begin{proof}
It only remains to show that $X \to L_E X$ is an $E$-equivalence. By the [[tensor triangulated category|tensor triangulated]] structure on $Ho(Spectra)$ (\href{Introduction+to+Stable+homotopy+theory+--+1-2#TensorTriangulatedStructureOnStableHomotopyCategory}{prop.}) the [[smash product of spectra|smash product]] with $E$ preserves homotopy cofiber sequences, so that

\begin{displaymath}
E \wedge G_E X
   \longrightarrow
  E \wedge X
    \overset{E \wedge \eta_X}{\longrightarrow}
  E \wedge L_E X
    \longrightarrow
  E \wedge \Sigma G_E X
\end{displaymath}
is also a homotopy cofiber sequence. But now $E \wedge G_E X \simeq 0$ by prop. \ref{ELocalizationCofiber}, and so the axioms (\href{Introduction+to+Stable+homotopy+theory+--+1-1#CategoryWithCofiberSequences}{def.}) of the [[triangulated category|triangulated structure]] on $Ho(Spectra)$ (\href{Introduction+to+Stable+homotopy+theory+--+1-1#StableHomotopyCategoryIsTriangulated}{prop.}) imply that $E \wedge \eta$ is an isomorphism.

\end{proof}
\begin{remark}
\label{Acyclification}\hypertarget{Acyclification}{}
Hence where $L_E$ is traditionally called ``$E$-localization'', $G_E$ might be called ``$E$-acyclification'', though that terminology is not used commonly.

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

\hypertarget{localization_at_moore_spectra_of_abelian_groups}{}\subsubsection*{{Localization at Moore spectra of abelian groups}}\label{localization_at_moore_spectra_of_abelian_groups}

A basic special case of $E$-localization of spectra is given for $E = S A$ the [[Moore spectrum]] of an [[abelian group]] $A$ (\hyperlink{Bousfield79}{Bousfield 79, section 2}). For $A = \mathbb{Z}_{(p)}$ this is [[p-localization]] and for $A = \mathbb{F}_p$ this is [[p-completion]], see examples \ref{pLocalization} and \ref{pCompletion} below for more.

\begin{prop}
\label{}\hypertarget{}{}
For $A_1$ and $A_2$ two [[abelian groups]] then the following are equivalent

\begin{enumerate}%
\item the Bousfield localizations at their [[Moore spectra]] are equivalent

\begin{displaymath}
L_{S A_1} \simeq L_{S A_2}
  \,;
\end{displaymath}

\item $A_1$ and $A_2$ have the same \emph{type of acyclicity}, meaning that

\begin{enumerate}%
\item every [[prime number]] $p$ is invertible in $A_1$ precisely if it is in $A_2$;


\item $A_1$ is a [[torsion group]] precisely if $A_2$ is.



\end{enumerate}


\end{enumerate}
\end{prop}
(\hyperlink{Bousfield79}{Bousfield 79, prop. 2.3}) recalled e.g. in (\hyperlink{VanKoughnett13}{VanKoughnett 13, prop. 4.2}).

This means that given an abelian group $A$ then

\begin{itemize}%
\item either $A$ is not torsion, then

\begin{displaymath}
L_{S A} \simeq L_{S \mathbb{Z}[I^{-1}]}
  \,,
\end{displaymath}
where $I$ is the set of primes invertible in $A$ and $\mathbb{Z}[I^{-1}] \hookrightarrow \mathbb{Q}$ is the [[localization of a ring|localization]] at these primes of the [[integers]];


\item or $A$ is torsion, then

\begin{displaymath}
L_{S A }\simeq L_{S(\oplus_q \mathbb{F}_q )  }
 \,,
\end{displaymath}
where the [[direct sum]] is over all [[cyclic groups]] of order $q$, for $q$ a prime not invertible in $A$.



\end{itemize}
\hypertarget{relation_to_nilpotent_completion}{}\subsubsection*{{Relation to nilpotent completion}}\label{relation_to_nilpotent_completion}

Let $E$ be an [[E-∞ ring]] and let $X$ be any [[spectrum]]

\begin{remark}
\label{CanonicalMapFromELocalizationToTotalization}\hypertarget{CanonicalMapFromELocalizationToTotalization}{}
There is a canonical map

\begin{displaymath}
L_E X \stackrel{}{\longrightarrow} \underset{\leftarrow}{\lim}_n (E^{\wedge^{n+1}_S}\wedge_S X)
\end{displaymath}
from the $E$-[[Bousfield localization of spectra]] of $X$ into the [[totalization]] of the canonical [[cosimplicial object|cosimplicial]] [[spectrum]] (see at \emph{[[nilpotent completion]]}).

\end{remark}
We now consider conditions for this morphism to be an [[equivalence]].

\begin{defn}
\label{CoreOfARing}\hypertarget{CoreOfARing}{}
For $R$ a [[ring]], its \emph{core} $c R$ is the [[equalizer]] in

\begin{displaymath}
c R 
    \longrightarrow 
  R 
    \stackrel{\longrightarrow}{\longrightarrow}
  R \otimes R
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{SufficientConditionsForTotalizationToBeELocalization}\hypertarget{SufficientConditionsForTotalizationToBeELocalization}{}
Let $E$ be a [[connective spectrum|connective]] [[E-∞ ring]] such that the core of $\pi_0(E)$, def. \ref{CoreOfARing}, is either of

\begin{itemize}%
\item the [[localization of a ring|localization]] of the [[integers]] at a set $J$ of [[primes]], $c \pi_0(E) \simeq \mathbb{Z}[J^{-1}]$;


\item $\mathbb{Z}_n$ for $n \geq 2$.



\end{itemize}
Then the map in remark \ref{CanonicalMapFromELocalizationToTotalization} is an equivalence

\begin{displaymath}
L_E X \stackrel{\simeq}{\longrightarrow} 
  \underset{\leftarrow}{\lim}_n (E^{\wedge^{n+1}_S}\wedge_S X)
  \,.
\end{displaymath}
\end{prop}
(\hyperlink{Bousfield79}{Bousfield 79}) see also for instance (\hyperlink{Bauer11}{Bauer 11, p.2})

For more discussion of [[E-infinity geometry|E-infinity]] (derived) [[formal completions]] via totalizations of [[Amitsur complexes]], see (\href{completion+of+a+module#Carlsson07}{Carlsson 07}).

\hypertarget{fracture_theorem}{}\subsubsection*{{Fracture theorem}}\label{fracture_theorem}

The [[fracture theorem]] says how Bousfield localization at a [[coproduct]]/[[wedge sum]] of spectra is a [[homotopy pullback]] of Bousfield localization separately. See at \emph{[[fracture theorem]]} for more on this.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

\begin{example}
\label{EModulesAreELocal}\hypertarget{EModulesAreELocal}{}
For $E$ an [[E-∞ ring]], every [[∞-module]] $X$ over $E$ is $E$-local, def. \ref{EAcyclicAndLocalSpectra}.

\end{example}
(e.g. \hyperlink{Lurie10}{Lurie, Lecture 20, example 5})

\begin{example}
\label{}\hypertarget{}{}
For $E$ an [[E-∞ algebra]] over an [[E-∞ ring]] $S$ and for $X$ an $S$-[[∞-module]], consider the dual [[Cech nerve]] [[cosimplicial object]]

\begin{displaymath}
E^{\wedge_S^{\bullet+1}}\wedge_S X \;\colon\; \Delta \longrightarrow Spectra
  \,.
\end{displaymath}
By example \ref{EModulesAreELocal} each term is $E$-local, so that the map to the [[totalization]]

\begin{displaymath}
X \longrightarrow \underset{\leftarrow}{\lim} E^{\wedge_S^{\bullet+1}} \wedge_S X
\end{displaymath}
factors through the $E$-localization of $X$

\begin{displaymath}
X \longrightarrow 
  L_E X
  \longrightarrow
  \underset{\leftarrow}{\lim} E^{\wedge_S^{\bullet+1}} \wedge_S X
  \,.
\end{displaymath}
Under suitable condition the second map here is indeed an [[equivalence]], in which case the [[totalization]] of the dual [[Cech nerve]] exhibits the $E$-localization. This happens for instance in the discussion of the [[Adams spectral sequence]], see the examples given there.

(see also e.g. \hyperlink{Bauer11}{Bauer 11, p. 2})

\end{example}
\hypertarget{rationalization}{}\subsubsection*{{Rationalization}}\label{rationalization}

\begin{example}
\label{Rationalization}\hypertarget{Rationalization}{}
Bousfield localization at the [[Moore spectrum]]/[[Eilenberg-MacLane spectrum]] $S \mathbb{Q}\simeq H\mathbb{Q}$ of the [[rational numbers]] is [[rationalization]] to [[rational homotopy theory]].

The corresponding $\mathbb{Q}$-acyclification (remark \ref{Acyclification}) is \emph{[[torsion approximation]]}.

\end{example}
e.g. (\hyperlink{Bauer11}{Bauer 11, example 1.7})

\hypertarget{localization}{}\subsubsection*{{$p$-Localization}}\label{localization}

For $p$ a [[prime number]] write $\mathbb{Z}_{(p)}$ for the [[localization of a ring|localization]] of the [[integers]] \emph{at} $(p)$, for the ring of integers [[localization of a ring|localized at]] $p$, hence with all primes \emph{except} $p$ inverted; equivalently the subring of the [[rational numbers]] with denominator not divisible by $p$.

\begin{example}
\label{pLocalization}\hypertarget{pLocalization}{}
The Bousfield localization at the [[Moore spectrum]] $S \mathbb{Z}_{(p)}$ is [[p-localization]].

\end{example}
(\hyperlink{Bousfield79}{Bousfield 79}), \hyperlink{Bauer11}{Bauer 11, example 1.7}). See at \emph{[[localization of a space]]} for details on this.

\begin{prop}
\label{pLocalizationIsSmashing}\hypertarget{pLocalizationIsSmashing}{}
$p$-localization is a [[smashing localization]]:

\begin{displaymath}
L_{S \mathbb{Z}_{(p)}} X
  \simeq
  S \mathbb{Z}_{(p)} \wedge X
  \,.
\end{displaymath}
\end{prop}
(\hyperlink{Bousfield79}{Bousfield 79, prop. 2.4}) recalled e.g. as (\hyperlink{VanKoughnett13}{van Koughnett 13, prop. 4.3}).

\hypertarget{completion}{}\subsubsection*{{$p$-Completion}}\label{completion}

For $p \in \mathbb{N}$ a [[prime number]], write

\begin{displaymath}
\mathbb{F}_p = \mathbb{Z}/(p)
\end{displaymath}
for the [[cyclic group]]/[[finite field]] of [[order of a group|order]] $p$.

\begin{defn}
\label{}\hypertarget{}{}
Write

\begin{displaymath}
\mathbb{Z}/p^\infty \coloneqq (\mathbb{Z}[p^{-1}])/\mathbb{Z}
\end{displaymath}
for the [[localization of a ring|localization]] of the [[integers]] away from $p$ followed by the quotient by $\mathbb{Z}$.

\end{defn}
e.g. (\hyperlink{Bousfield79}{Bousfield 79, p. 6})

\begin{remark}
\label{}\hypertarget{}{}
The [[short exact sequence]] of [[abelian groups]]

\begin{displaymath}
0 \to \mathbb{Z} \longrightarrow \mathbb{Z}[p^{-1}] \longrightarrow \mathbb{Z}/p^\infty \to 0
\end{displaymath}
induces the [[homotopy fiber sequence]] (in [[spectra]]) of [[Moore spectra]]

\begin{displaymath}
\Omega S(\mathbb{Z}/p^\infty) \longrightarrow S\mathbb{Z} \longrightarrow S(\mathbb{Z}[p^{-1}])
 \,.
\end{displaymath}
As in \hyperlink{Bousfield79}{Bousfield 79, p. 6} one also writes

\begin{displaymath}
S^{-1} (\mathbb{Z}/p^\infty)
  \coloneqq 
  \Omega S(\mathbb{Z}/p^\infty)
  \,.
\end{displaymath}
\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
The localization of spectra at the [[Moore spectrum]] $S\mathbb{F}_p$ is given by the [[mapping spectrum]] out of $S^{-1} \mathbb{Z}/p^\infty$:

\begin{displaymath}
L_{S \mathbb{F}_p} X
  \simeq
  [\Omega S \mathbb{Z}/p^\infty, X]
  \,.
\end{displaymath}
\end{prop}
(\hyperlink{Bousfield79}{Bousfield 79, prop. 2.5})

Fact: $\mathbb{F}_p$-localizaton is [[p-completion]], e.g. \hyperlink{LurieProper}{Lurie ``Proper Morphisms\ldots{}'', section 4}.

\begin{example}
\label{}\hypertarget{}{}
Let

\begin{displaymath}
E \coloneqq H \mathbb{Z}/p\mathbb{Z}
\end{displaymath}
be the corresponding [[Moore spectrum]]. Then a spectrum which corresponds to a [[chain complex]] under the \href{module+spectrum#StableDoldKanCorrespondence}{stable Dold-Kan corespondence} is $E$-local, def. \ref{AcyclicAndLocal}, if that chain complex has [[chain homology]] groups being $\mathbb{Z}[p^{-1}]$-modules.

The $E$-localization of a spectrum in this case is \emph{[[p-completion]]}.

\end{example}
(e.g. \hyperlink{Lurie10}{Lurie, Lecture 20, example 8})

More generally

\begin{example}
\label{pCompletion}\hypertarget{pCompletion}{}
Bousfield localization at the [[Moore spectrum]] $S \mathbb{F}_p$ is [[p-completion]] to [[p-adic homotopy theory]].

\end{example}
E.g. (\hyperlink{Bauer11}{Bauer 11, example 1.7}). See at \emph{[[localization of a space]]} for more on this.

\hypertarget{telescopic_localization}{}\subsubsection*{{Telescopic localization}}\label{telescopic_localization}

\begin{itemize}%
\item [[telescopic localization]]

\end{itemize}
\hypertarget{chromatic_localization}{}\subsubsection*{{Chromatic localization}}\label{chromatic_localization}

\begin{itemize}%
\item [[chromatic localization]]

\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[smashing localization]]


\item [[chromatic filtration]], [[chromatic layer]]


\item [[K(n)-local stable homotopy theory]]



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

Original articles are

\begin{itemize}%
\item [[Aldridge Bousfield]], \emph{The localization of spectra with respect to homology} , Topology vol 18 (1979) (\href{http://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/bousfield-topology-1979.pdf}{pdf})


\item [[Douglas Ravenel]], \emph{Localization with respect to certain periodic homology theories}, American Journal of Mathematics, Vol. 106, No. 2, (Apr., 1984), pp. 351-414 (\href{https://www.math.rochester.edu/people/faculty/doug/mypapers/loc.pdf}{pdf})



\end{itemize}
Discussion in terms of [[Bousfield localization of model categories]] [[model category of spectra|of spectra]] appears in

\begin{itemize}%
\item [[Anthony Elmendorf]], [[Igor Kriz]], [[Michael Mandell]], [[Peter May]], chapter VIII of \emph{[[Rings, modules and algebras in stable homotopy theory]]} 1997 (\href{http://www.math.uchicago.edu/~may/BOOKS/EKMM.pdf}{pdf})


\item [[Stefan Schwede]], [[Brooke Shipley]], lemma 4.1 in \emph{A uniqueness theorem for stable homotopy theory}, Mathematische Zeitschrift 239 (2002), 803-828 (\href{https://arxiv.org/abs/math/0012021}{arXiv:math/0012021})



\end{itemize}
see also

\begin{itemize}%
\item [[Carles Casacuberta]], [[Javier Gutiérrez]], \emph{Homotopical Localizations of Module Spectra}, Transactions of the American Mathematical Society Vol. 357, No. 7 (Jul., 2005), pp. 2753-2770 (\href{http://www.jstor.org/stable/3845180}{jstor})

\end{itemize}
Lecture notes include

\begin{itemize}%
\item [[Nerses Aramian]], \emph{Bousfield Localization} (\href{http://www.math.uiuc.edu/~aramyan2/bousfield.pdf}{pdf})


\item [[Tilman Bauer]], \emph{Bousfield localization and the Hasse square}, 2011 (\href{http://math.mit.edu/conferences/talbot/2007/tmfproc/Chapter09/bauer.pdf}{pdf})


\item [[Paul VanKoughnett]], \emph{Spectra and localization}, 2013 ([[VanKoughnettLocalization.pdf:file]])



\end{itemize}
Discussion the general context of [[higher algebra]]/[[stable homotopy theory]] includes

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Chromatic Homotopy Theory]]}, Lecture notes (2010) (\href{http://www.math.harvard.edu/~lurie/252x.html}{web}), Lecture 20 \emph{Bousfield localization} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture20.pdf}{pdf})


\item [[Jacob Lurie]], section 4 of \emph{[[Proper Morphisms, Completions, and the Grothendieck Existence Theorem]]}



\end{itemize}
Discussion specifically of [[K(n)-local spectra]] includes

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

\end{itemize}
See also

section 2.4 of

\begin{itemize}%
\item Holger Reeker, \emph{On K(1)-local SU-bordism} (\href{http://arxiv.org/abs/0907.4299}{arXiv:0907.4299})

\end{itemize}
[[!redirects local spectrum]] [[!redirects local spectra]]

[[!redirects localization of a spectrum]] [[!redirects localization of spectra]]

[[!redirects localization of spectra]] [[!redirects localizations of spectra]]



\end{document}