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\begin{document}

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\section*{localization of a module}

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

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{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{for_modules_over_rings}{For modules over rings}\dotfill \pageref*{for_modules_over_rings} \linebreak
\noindent\hyperlink{for_chain_complexes}{For chain complexes}\dotfill \pageref*{for_chain_complexes} \linebreak
\noindent\hyperlink{ForSpectra}{For spectra ($\mathbb{S}$-modules)}\dotfill \pageref*{ForSpectra} \linebreak
\noindent\hyperlink{for_modules_over_rings_2}{For $\infty$-modules over $E_\infty$-rings}\dotfill \pageref*{for_modules_over_rings_2} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{eilenbergwatts_theorem}{Eilenberg-Watts theorem}\dotfill \pageref*{eilenbergwatts_theorem} \linebreak
\noindent\hyperlink{relation_with_torsion_approximation_and_completion}{Relation with torsion approximation and completion}\dotfill \pageref*{relation_with_torsion_approximation_and_completion} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{literature}{Literature}\dotfill \pageref*{literature} \linebreak


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

A \emph{localization of a module} is the result of application of an [[additive functor|additive]] [[localization functor]] on a [[category of modules]] over some [[ring]] $R$.

When $R$ is a [[commutative ring]] of functions, and under the \href{module#RelationToVectorBundlesInIntroduction}{interpretation of modules as generalized vector bundles} the \emph{localization} of a module corresponds to the restriction of the bundle to a subspace of its base space.

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

\hypertarget{for_modules_over_rings}{}\subsubsection*{{For modules over rings}}\label{for_modules_over_rings}

For $R$ a (possibly noncommutative) unital ring, let $\mathcal{A} = R$[[Mod]] be the [[category]] $R$-[[modules]]. Here $R$ may be the [[structure sheaf]] of some [[ringed topos]] and accordingly the modules may be [[sheaves of modules]].

Consider a [[reflective localization]] functor

\begin{displaymath}
Q^* = Q^*_\Sigma \colon \mathcal{A}\to \Sigma^{-1}\mathcal{A}
\end{displaymath}
with [[right adjoint]] $Q_*$. The application of this functor to a [[module]] $M\in \mathcal{A}$ is some object $Q^*(M)$ in the localized category $\Sigma^{-1}\mathcal{A}$, which is up to isomorphism determined by its image $Q_* Q^*(M)$.

The \textbf{localization map} is the component of the [[unit of an adjunction|unit of the adjunction]] (usually denoted by $i$, $j$ or $\iota$ in this setup) $\iota_M : M\to Q_* Q^*(M)$.

Depending on an author or a context, we say that a (reflective) localization functor of category of modules is \textbf{flat} if either $Q^*$ is also left [[exact functor]], or more strongly that the composed endofunctor $Q_* Q^*$ is exact. For example, [[Gabriel localization]] is flat in the first, weak sense, and left or right Ore localization is flat in the second, stronger, sense.

\hypertarget{for_chain_complexes}{}\subsubsection*{{For chain complexes}}\label{for_chain_complexes}

\ldots{} Greenlees-May duality\ldots{}

\hypertarget{ForSpectra}{}\subsubsection*{{For spectra ($\mathbb{S}$-modules)}}\label{ForSpectra}

\begin{prop}
\label{}\hypertarget{}{}
Suppose that $L \colon Spectra \to Spectra$ is a [[smashing localization]] given by [[smash product]] with some [[spectrum]] $T$. Write $F$ for the [[homotopy fiber]]

\begin{displaymath}
F \longrightarrow \mathbb{S} \longrightarrow T
  \,.
\end{displaymath}
Then there is a [[fracture diagram]] of operations

\begin{displaymath}
\itexarray{
    T \wedge (-) && \longleftarrow  &&  [T,-]
    \\
    & \nwarrow && \swarrow
    \\
    && \mathbb{S}
    \\
    & \swarrow & & \nwarrow
    \\
    [F,-] && \longleftarrow && F \wedge (-)
  }
\end{displaymath}
where $[F,-]$ and $T \wedge (-) \colon Spectr \to Spectra$ are [[idempotent (∞,1)-monads]] and $[T,-]$, $[F,-]$ are idempotent $\infty$-comonad, the diagonals are [[homotopy fiber sequences]].

\end{prop}
([[Charles Rezk]], \href{http://mathoverflow.net/a/178316/381}{MO comment,August 2014})

\begin{example}
\label{}\hypertarget{}{}
For $T = S \mathbb{Z}[p^{-1}]$ the [[Moore spectrum]] of the [[integers]] [[localization of a ring|localized away from]] $p$, then

\begin{displaymath}
F = \Sigma^{-1} S (\mathbb{Z}[p^{-1}]/\mathbb{Z})
   \to 
  \mathbb{S}
   \to
  S \mathbb{Z}[p^{-1}]
\end{displaymath}
and hence

\begin{itemize}%
\item $\Sigma^{-1} S (\mathbb{Z}[p^{-1}]/\mathbb{Z}) \wedge (-)$ is $p$-[[torsion approximation]];


\item $[\Sigma^{-1} S (\mathbb{Z}[p^{-1}]/\mathbb{Z}),-]$ is $p$-[[completion of a module|completion]];


\item $S \mathbb{Z}[p^{-1}] \wedge (-)$ is localization away from $p$ ($p$-[[rationalization]])


\item $[T,-]$ is forming $p$-[[adic residual]].



\end{itemize}
\begin{displaymath}
\itexarray{
    &&  localization\;away\;from\;\mathfrak{a} && \stackrel{}{\longrightarrow} && \mathfrak{a}\;adic\;residual
    \\
    & \nearrow & & \searrow & & \nearrow && \searrow
    \\
    && && X &&  &&  
    \\
    & \searrow &  & \nearrow & & \searrow && \nearrow
    \\
    && formal\;completion\;at\;\mathfrak{a}\; && \longrightarrow && \mathfrak{a}\;torsion\;approximation
  }
  \,,
\end{displaymath}
\end{example}
With (\hyperlink{Bousfield79}{Bousfield 79, prop.2.5})

\hypertarget{for_modules_over_rings_2}{}\subsubsection*{{For $\infty$-modules over $E_\infty$-rings}}\label{for_modules_over_rings_2}

Let $A$ be an [[E-∞ ring]] and $\mathfrak{a} \subset \pi_0 A$ a [[generators and relations|finitely generated]] ideal of its underlying [[commutative ring]].

\begin{defn}
\label{LocalInfinityModule}\hypertarget{LocalInfinityModule}{}
An $A$-[[∞-module]] $N$ is an \emph{$\mathfrak{a}$-local module} if for every $\mathfrak{a}$-[[torsion module]] $T$ (def. \ref{TorsionInfinityModule}), the [[derived hom space]]

\begin{displaymath}
Hom_A(T,N) \simeq \ast
\end{displaymath}
is contractible.

\end{defn}
(\hyperlink{LurieCompletions}{Lurie ``Completions'', def. 4.1.9}).

\begin{prop}
\label{LocalizationAwayByColimit}\hypertarget{LocalizationAwayByColimit}{}
For $\mathfrak{a} = (a)$ generated from a single element, then the \href{localization+of+a+commutative+ring#ForEInfinityRings}{localization of an (∞,1)-ring}-map $A \to A[a^{-1}]$ is given by the [[(∞,1)-colimit]] over the sequence of right-multiplication with $a$

\begin{displaymath}
A[a^{-1}]
  \simeq
  \underset{\rightarrow}{\lim}
  (
    A \stackrel{\cdot a}{\longrightarrow} A
    \stackrel{\cdot a}{\longrightarrow} A
    \stackrel{\cdot a}{\longrightarrow}  \cdots
  )
  \,.
\end{displaymath}
\end{prop}
(\hyperlink{LurieCompletions}{Lurie ``Completions'', remark 4.1.11})

\begin{prop}
\label{}\hypertarget{}{}
The [[full sub-(∞,1)-category]]

\begin{displaymath}
A Mod_{\mathfrak{a}loc}
  \hookrightarrow
  A Mod
\end{displaymath}
of [[∞-modules]] [[localization of a module|local]] away from $\mathfrak{a}$ is [[reflective sub-(∞,1)-category|reflective]]. The reflector

\begin{displaymath}
\Pi_{\mathfrak{a}dR} \colon A Mod \longrightarrow A Mod_{\mathfrak{a}loc}
\end{displaymath}
is called \emph{localization}.

\end{prop}
\begin{prop}
\label{}\hypertarget{}{}
There is a [[natural transformation|natural]] [[homotopy fiber sequence]]

\begin{displaymath}
&#643;_{\mathfrak{a}} \longrightarrow id \longrightarrow &#643;_{\mathfrak{a}dR}
\end{displaymath}
relating $\mathfrak{a}$-[[torsion approximation]] on the left with $\mathfrak{a}$-localization on the right.

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

\hypertarget{eilenbergwatts_theorem}{}\subsubsection*{{Eilenberg-Watts theorem}}\label{eilenbergwatts_theorem}

By the [[Eilenberg-Watts theorem]], if $\mathcal{A}= R$[[Mod]] then the localization of a module

\begin{displaymath}
Q^*(M) = Q^*(R)\otimes_R M
\end{displaymath}
is given by forming the [[tensor product of modules]] with the localizatin of the ring $R$, regarded as a module over itself.

If the localization is a left [[Ore localization]] or [[commutative localization]] at a set $S\subset R$ then $Q^*(R) = S^{-1} R$ is the [[localization of a ring|localization of the ring]] itself and hence in this case the localization of the module

\begin{displaymath}
Q^*(M) = S^{-1} R\otimes_R M
\end{displaymath}
is given by [[extension of scalars]] along the localization map $R \to S^{-1}R$ of the ring itself.

In these cases there are also direct constructions of $Q^*(M)$ (not using to $Q^*(R)$) which give an isomorphic result, also denoted by $S^{-1}M$.

\hypertarget{relation_with_torsion_approximation_and_completion}{}\subsubsection*{{Relation with torsion approximation and completion}}\label{relation_with_torsion_approximation_and_completion}

[[!include arithmetic cohesion -- table]]

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

\begin{itemize}%
\item [[localization of an abelian group]]


\item [[localization of a ring]], [[Ore localization]], [[Gabriel localization]], [[Cohn localization]]


\item [[localization of a space]] (and of a [[spectrum]])


\item [[localization of a category]] (= localization functor)


\item [[locally free module]]



\end{itemize}
\hypertarget{literature}{}\subsection*{{Literature}}\label{literature}

Standard discussion over [[commutative rings]] is for instance in

\begin{itemize}%
\item Andreas Gathmann, \emph{Localization} ([[GathmannLocalization.pdf:file]])

\end{itemize}
Discussion in the general case of [[noncommutative geometry]] is in

\begin{itemize}%
\item [[Z. ?koda]], \emph{Noncommutative localization in noncommutative geometry}, London Math. Society Lecture Note Series 330 (\href{http://www.maths.ed.ac.uk/~aar/books/nlat.pdf}{pdf}), ed. A. Ranicki; pp. 220--313, \href{http://arxiv.org/abs/math.QA/0403276}{math.QA/0403276}

\end{itemize}
Discussion in the context of [[spectra]] originates in

\begin{itemize}%
\item [[Aldridge Bousfield]], [[Daniel Kan]], \emph{[[Homotopy limits, completions and localizations]]}, Lecture Notes in Mathematics, Vol 304, Springer 1972


\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 [[Dennis Sullivan]], \emph{Geometric topology: localization, periodicity and Galois symmetry}, volume 8 of K- Monographs in Mathematics. Springer, Dordrecht, 2005. The 1970 MIT notes, Edited and with a preface by [[Andrew Ranicki]] (\href{http://www.maths.ed.ac.uk/~aar/books/gtop.pdf}{pdf})



\end{itemize}
Discussion in the context of [[higher algebra]] is in

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

\end{itemize}
[[!redirects localizations of a module]] [[!redirects localizations of modules]]



\end{document}