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\begin{document}

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\section*{completion of a module}

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

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

\hypertarget{formal_geometry}{}\paragraph*{{Formal geometry}}\label{formal_geometry}

[[!include formal geometry -- 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_a_commutative_ring}{For modules over a commutative ring}\dotfill \pageref*{for_modules_over_a_commutative_ring} \linebreak
\noindent\hyperlink{for_modules_over_an_ring}{For $\infty$-modules over an $E_\infty$-ring}\dotfill \pageref*{for_modules_over_an_ring} \linebreak
\noindent\hyperlink{TorsionApproximation}{Torsion approximation}\dotfill \pageref*{TorsionApproximation} \linebreak
\noindent\hyperlink{Localization}{Localization}\dotfill \pageref*{Localization} \linebreak
\noindent\hyperlink{completion}{Completion}\dotfill \pageref*{completion} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{general_properties}{General properties}\dotfill \pageref*{general_properties} \linebreak
\noindent\hyperlink{Monoidalness}{Monoidalness}\dotfill \pageref*{Monoidalness} \linebreak
\noindent\hyperlink{relation_to_localization}{Relation to localization}\dotfill \pageref*{relation_to_localization} \linebreak
\noindent\hyperlink{RelationToTorsionApproximation}{Relation of formal completion to torsion approximation}\dotfill \pageref*{RelationToTorsionApproximation} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Directly analogous to the concept of [[completion of a ring]] is the \emph{completion of a module} over that ring.

In particular the [[formal completion]] or \emph{adic completion} of a ring $A$ at an ideal $\mathfrak{a}$ has a corresponding analog for modules. Where the adic completion $A_{\mathfrak{a}}^\wedge$ of the ring itself has the geometric interpretation of forming the [[formal neighbourhood]] $Spf(A_{\mathfrak{a}}^\wedge)$ of [[spectrum of a commutative ring|ring spectra]] $Spec(A/\mathfrak{a}) \hookrightarrow Spec(A)$, so under the interpretation (see \href{module#RelationToVectorBundlesInIntroduction}{here}) of $A$-[[modules]] as [[bundles]] over $Spec(A)$, the $\mathfrak{a}$-adic completion $N_{\mathfrak{a}}^\wedge$ of an $A$-module $N$ has the interpretation of being the restriction of that bundle to that formal neighbourhood.

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

\hypertarget{for_modules_over_a_commutative_ring}{}\subsubsection*{{For modules over a commutative ring}}\label{for_modules_over_a_commutative_ring}

\begin{defn}
\label{TraditionalDefinition}\hypertarget{TraditionalDefinition}{}
For $A$ a [[commutative ring]], $\mathfrak{a} \subset A$ an ideal in $A$ and for $N$ an $A$-[[module]], then the \emph{$\mathfrak{a}$-adic completion} or \emph{formal completion at $\mathfrak{a}$} of $N$ is the [[filtered limit]]

\begin{displaymath}
N^{\wedge}_{\mathfrak{a}}
  \coloneqq
  \underset{\leftarrow}{\lim}_n
  N/(\mathfrak{a}^n N)
\end{displaymath}
of [[quotients]] of $N$ by the submodules induced by all powers of the ideal.

\end{defn}
There is a canonical projection map $N \longrightarrow N^\wedge_{\mathfrak{a}}$. Its [[kernel]] is sometimes called the $\mathfrak{a}$-[[adic residual]].

\hypertarget{for_modules_over_an_ring}{}\subsubsection*{{For $\infty$-modules over an $E_\infty$-ring}}\label{for_modules_over_an_ring}

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]].

\hypertarget{TorsionApproximation}{}\paragraph*{{Torsion approximation}}\label{TorsionApproximation}

\begin{defn}
\label{TorsionInfinityModule}\hypertarget{TorsionInfinityModule}{}
An $A$-[[∞-module]] $N$ is an \emph{$\mathfrak{a}$-torsion module} if for all elements $n \in \pi_k N$ and all elements $a \in \mathfrak{a}$ there is $k \in \mathbb{N}$ such that $a^k n = 0$.

\end{defn}
(\hyperlink{LurieCompletions}{Lurie ``Completions'', def. 4.1.3}).

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

\begin{displaymath}
A Mod_{\mathfrak{a}tor}
  \hookrightarrow
  A Mod
\end{displaymath}
is [[reflective sub-(∞,1)-category|co-reflective]] and the co-reflector $\Pi_{\mathfrak{a}}$ -- the \emph{[[torsion approximation]]} -- is [[smashing localization|smashing]].

\end{prop}
(\hyperlink{LurieCompletions}{Lurie ``Completions'', prop. 4.1.12}).

\begin{prop}
\label{}\hypertarget{}{}
For $N \in A Mod_{\leq 0}$ then [[torsion approximation]], prop. \ref{TorsionApproximation}, intuced a [[monomorphism]] on $\pi_0$

\begin{displaymath}
\pi_0 \Pi_{\mathfrak{a}} N \hookrightarrow \pi_0 N
\end{displaymath}
including the $\mathfrak{a}$-nilpotent elements of $\pi_0 N$.

\end{prop}
(\hyperlink{LurieCompletions}{Lurie ``Completions'', prop. 4.1.18}).

\hypertarget{Localization}{}\paragraph*{{Localization}}\label{Localization}

\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 of a module|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 of a module|localization]] on the right.

\end{prop}
\hypertarget{completion}{}\paragraph*{{Completion}}\label{completion}

\begin{defn}
\label{InfinityCompletion}\hypertarget{InfinityCompletion}{}
An [[∞-module]] $N$ over $A$ is \emph{$\mathfrak{a}$-complete} if for all $\mathfrak{a}$-local $\infty$-modules $L$ (def. \ref{LocalInfinityModule}) then $Hom_A(L,N) \simeq \ast$.

The [[full sub-(∞,1)-category]]

\begin{displaymath}
A Mod_{\mathfrak{a}comp}
  \hookrightarrow
  A Mod
\end{displaymath}
of the [[(∞,1)-category of ∞-modules]] on the $\mathbb{a}$-complete ones is a [[reflective sub-(∞,1)-category]]. The reflector

\begin{displaymath}
\flat_{\mathfrak{a}} \colon A Mod \longrightarrow A Mod_{\mathfrak{a}comp} \hookrightarrow A Mod
\end{displaymath}
\begin{displaymath}
N \mapsto N^{\wedge}_{\mathfrak{a}}
\end{displaymath}
is called \emph{$\mathfrak{a}$-completion}.

\end{defn}
(\hyperlink{LurieCompletions}{Lurie ``Completions'', def. 4.2.1, lemma 4.2.2}).

Definition \ref{InfinityCompletion} relates to the traditional definition, def. \ref{TraditionalDefinition}, as follows

\begin{prop}
\label{}\hypertarget{}{}
Let $N$ a homotopically discrete [[∞-module]] over the [[E-∞ ring]] $A$ which is a [[Noetherian module]] in that all its submodules are finitely [[generators and relations|finitely generated]]. Then the $\mathfrak{a}$-completion of $N$ in the sense of def. \ref{InfinityCompletion} coincides with the traditional definition def. \ref{TraditionalDefinition}.

\end{prop}
(\hyperlink{LurieCompletions}{Lurie ``Completions'', prop. 4.3.6})

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

\hypertarget{general_properties}{}\subsubsection*{{General properties}}\label{general_properties}

The [[full sub-(∞,1)-category]] $A Mod_{\mathfrak{a} comp}$ is a [[locally presentable (∞,1)-category]].

(\hyperlink{LurieCompletions}{Lurie ``Completions'', prop. 4.1.17})

\hypertarget{Monoidalness}{}\subsubsection*{{Monoidalness}}\label{Monoidalness}

We discuss how both $\mathfrak{a}$-completion $\flat_{\mathfrak{a}}$ and $\mathfrak{a}$-[[torsion approximation]] $\Pi_{\mathfrak{a}}$ on $A Mod$ are [[monoidal (∞,1)-functors]] with respect to the [[smash product of spectra]] over $A$.

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{prop}
\label{}\hypertarget{}{}
The completion reflection $\flat_{\mathfrak{a}}$, def. \ref{InfinityCompletion}, is a [[monoidal (∞,1)-functor]].

\end{prop}
(\hyperlink{LurieCompletions}{Lurie ``Completions'', remark 4.2.6}).

For the torsion approximation functor $\Pi_{\mathfrak{a}}$ one gets something slightly weaker, it preserves ``monoids without unit'':

\begin{prop}
\label{}\hypertarget{}{}
The [[full sub-(∞,1)-category]] of $\mathfrak{a}$-torsion modules, def. \ref{TorsionInfinityModule}, is [[reflective sub-(∞,1)-category|co-reflective]]

\begin{displaymath}
A Mod_{\mathfrak{a}tor}
  \stackrel{\hookrightarrow}{\underset{\Pi_{\mathfrak{a}}}{\longleftarrow}}
  A Mod
 \,.
\end{displaymath}
Moreover, the coreflector $\Pi_{\mathfrak{a}}$ is ``[[smashing localization|smashing]]'', in that there is $V \in A Mod$ such that $\Pi_{\mathfrak{a}}(-) \simeq V \wedge (-)$ is given by the [[smash product]] with $V$. If $\mathfrak{a} = (\{x_i\}_i)$ then $V$ is the [[tensor product]] $V =\underset{i}{\otimes} V_i$ over all the [[homotopy fibers]]

\begin{displaymath}
\Omega (A[x_i^{-1}]/A) \longrightarrow A \longrightarrow A[x_i^{-1}]
  \,.
\end{displaymath}
\end{prop}
(\hyperlink{LurieCompletions}{Lurie ``Completions'', prop. 4.1.12}).

From the general properties of [[smashing localization]] it follows that

\begin{cor}
\label{}\hypertarget{}{}
The coreflection $\Pi_{\mathfrak{a}} \colon A Mod \to A Mod$

\begin{enumerate}%
\item preserves small [[(∞,1)-colimits]];


\item is a ``[[monoidal (∞,1)-functor]]'' except possibly for preservation of units.



\end{enumerate}
\end{cor}
See also (\hyperlink{LurieCompletions}{Lurie ``Completions'', cor. 4.1.16}).

\hypertarget{relation_to_localization}{}\subsubsection*{{Relation to localization}}\label{relation_to_localization}

The [[homotopy cofiber]] of $\mathfrak{a}$-completion $\Pi_{\mathfrak{a}}$ is [[localization of a module|localization]] away from $\mathfrak{a}$, in that there is a [[homotopy fiber sequence]]

\begin{displaymath}
(-)_{\mathfrak{a}}^{\wedge} \to id \to (-)[\mathfrak{a}^{-1}]
\end{displaymath}
with the completion functor of def. \ref{InfinityCompletion} on the left and the localization functor of prop. \ref{LocalizationAwayByColimit} on the right.

(\hyperlink{LurieCompletions}{Lurie ``Completions'', example 4.1.14, remark 4.1.20})

\hypertarget{RelationToTorsionApproximation}{}\subsubsection*{{Relation of formal completion to torsion approximation}}\label{RelationToTorsionApproximation}

For suitable ideals $\mathfrak{a}\subset A$ of a commutative ring $A$ or more generally of an [[E-∞ ring]], then the [[derived functor]] of $\mathfrak{a}$-adic [[completion of a module|completion of A-modules]] forms together with $\mathfrak{a}$-[[torsion approximation]] an [[adjoint modality]] on the\newline [[(∞,1)-category of modules]] over $A$. See at \emph{[[fracture square]]} for details.

[[!include arithmetic cohesion -- table]]

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

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}
Discussion of formal completion of [[(infinity,1)-modules]] in terms of [[totalization]] of [[Amitsur complexes]] is in

\begin{itemize}%
\item [[Gunnar Carlsson]], \emph{Derived completions in stable homotopy theory}, Journal of Pure and Applied Algebra Volume 212, Issue 3, March 2008, Pages 550--577 (\href{http://arxiv.org/abs/0707.2585}{arXiv:0707.2585})

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

[[!redirects completion of modules]] [[!redirects completions of modules]]

[[!redirects formal completion of a module]]



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