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\begin{document}

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\section*{completion of a ring}

\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{ideas}{Ideas}\dotfill \pageref*{ideas} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{formaladic_completion_and_formal_neighbourhoods}{Formal/adic completion and formal neighbourhoods}\dotfill \pageref*{formaladic_completion_and_formal_neighbourhoods} \linebreak
\noindent\hyperlink{derived_completion_functor}{Derived completion functor}\dotfill \pageref*{derived_completion_functor} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{RelationToTorsionApproximation}{Relation of formal completion to torsion approximation}\dotfill \pageref*{RelationToTorsionApproximation} \linebreak
\noindent\hyperlink{as_modality_in_arithmetic_cohesion}{As modality in arithmetic cohesion}\dotfill \pageref*{as_modality_in_arithmetic_cohesion} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{ideas}{}\subsection*{{Ideas}}\label{ideas}

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

Quite generally, the \emph{completion of a ring} is a [[completion]] of a [[topological ring]] to a [[complete space|complete]] [[topological ring]], when possible, for instance of a [[normed ring]] to a [[Banach ring]].

\hypertarget{formaladic_completion_and_formal_neighbourhoods}{}\subsubsection*{{Formal/adic completion and formal neighbourhoods}}\label{formaladic_completion_and_formal_neighbourhoods}

A special case of ring completion is the \emph{formal completion} or \emph{adic completion} of a [[commutative ring]] $R$, which is its topological completion with respect to the [[adic topology]] induced by a [[maximal ideal]] $I\subset R$ (\hyperlink{Sullivan05}{Sullivan 05, definition 1.3}).

The underlying ring $\widehat R_I$ of this formal completion is the [[limit]]

\begin{displaymath}
\widehat R_I \coloneqq \underset{\leftarrow}{\lim} (R/I^n)
\end{displaymath}
(formed in the [[category]] [[CRing]] of [[commutative rings]]) of the [[quotients]] of $R$ by all the [[powers]] of this ideal, $I$ (\hyperlink{Sullivan05}{Sullivan 05, proposition 1.13}). Notice that this may be considered purely algebraically.

In words, this limit construction says that the elements of $R_I$ are sequences of elements in $R$ which ``successively add smaller and smaller elements, as seen by the ideal $I$''. This is as for [[formal power series]] rings, which are indeed the archetypical example of formal completions, see example \ref{FormalPowerSeries} below.

Generally, the [[Isbell duality|dual]] [[geometry|geometric]] meaning of formal ring completion is in [[formal geometry]]: the proper [[spectrum (geometry)|geometric spectrum]] of a formally completed ring is known as a \emph{[[formal spectrum]]} $Spf(R,I)$. Geometrically this is the [[formal neighbourhood]] of the [[spectrum of a commutative ring|spectrum]] $Spec(R/I)$ inside $Spec(R)$.

\begin{displaymath}
Spec(R/I)
  \hookrightarrow
  Spf(\widehat R_I)
  \hookrightarrow
  Spec(R)
  \,.
\end{displaymath}
\hypertarget{derived_completion_functor}{}\subsubsection*{{Derived completion functor}}\label{derived_completion_functor}

The [[derived functor]] of adic completion was originally discussed in (\hyperlink{GreenleesMay92}{Greenless-May 92} (``Greenlees-May duality''). For discussion of its relation to derived [[torsion subgroup]] functor see (\hyperlink{PortaShaulYekutieli10}{Porta-Shaul-Yekutieli 10}) and see at \emph{\href{http://ncatlab.org/nlab/show/fracture+theorem#CompletionAndTorsionOnDerivedCategories}{fracture theorem -- Arithmetic fracturing for chain complexes}}.

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

The archetypical example which most clearly exhibits the [[geometry|geometric]] meaning of formal completions of rings is the following

\begin{example}
\label{FormalPowerSeries}\hypertarget{FormalPowerSeries}{}
For $R$ any [[ring]] and $R[x]$ the [[polynomial ring]] with [[coefficients]] in $R$, then the formal completion of $R[x]$ at the ideal $(x)$ generated by the free generator $x$ is the ring of [[formal power series]] $R[ [x] ]$.

If $R$ is a [[field]], then geometrically $Spec(R[x])$ is the [[affine line]] in [[algebraic geometry]]/[[arithmetic geometry]] over $R$, while $Spf(R[ [x] ])$ is the [[formal disk]] inside the affine line around the origin.

\end{example}
The key class of example of completions in [[non-archimedean analytic geometry]] is the following.

\begin{example}
\label{pAdicNumbers}\hypertarget{pAdicNumbers}{}
The [[p-adic integers]] are the completion of the ring of [[integers]] at the [[prime ideal]] $(p) \subset \mathbb{Z}$. Similarly the [[p-adic rational numbers]] are the completion of the [[rational numbers]] $(p)$, and the [[p-adic complex numbers]] are the completion of the [[complex numbers]] at $(p)$.

\end{example}
\begin{remark}
\label{}\hypertarget{}{}
In view of the example \ref{FormalPowerSeries} one sees that the $p$-adic numbers in \ref{pAdicNumbers} are in fact [[analogy|analogous]] to [[formal power series]] rings, hence that they behave like [[function rings]] on [[formal disks]] in \emph{some} kind of geometry.

This [[analogy]] is part of what is known as the [[function field analogy]], which says that the ring of [[integers]] $\mathbb{Z}$ behaves like the would-be ``ring of polynomials $\mathbb{F}_1[x]$ over [[F1]]''.

\end{remark}
\begin{example}
\label{AtiyahSegalTheorem}\hypertarget{AtiyahSegalTheorem}{}
The [[Atiyah-Segal completion theorem]] states that for $G$ a [[topological group]] and $X$ a [[G-space]], then the [[topological K-theory]] ring $K(X//G)$ of the [[homotopy quotient]] $X//G$ (of the [[Borel construction]]) is the completion of the $G$-[[equivariant K-theory]] ring of $X$.

So this says that the ``very naive'' equivariant K-theory embodied by $K(X//G)$ is an infinitesimal approximation to the genuine equivariant K-theory.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
The phenonemon of example \ref{AtiyahSegalTheorem} appears for other [[generalized cohomology theories]] too, such as [[complex cobordism]] (\hyperlink{GreenMay97}{GreenleesMay 97}). For [[complex oriented cohomology theories]] it says that the [[formal group]] assigned by these to $\ast//U(1)$ is to be thought of as the formal completion of a more globally defined something. One case where this ``something'' has been understood in some detail is that where the cohomology theory is [[elliptic cohomology]]. In that case the analog of equivariant K-theory is \emph{[[equivariant elliptic cohomology]]}, see there for more details.

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

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

For suitable ideals $\mathfrak{a}\subset A$ of a commutative ring $A$, 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{\href{fracture+theorem#CompletionAndTorsionOnDerivedCategories}{arithmetic fracturing for chain complexes}} for details.

\hypertarget{as_modality_in_arithmetic_cohesion}{}\subsubsection*{{As modality in arithmetic cohesion}}\label{as_modality_in_arithmetic_cohesion}

[[!include arithmetic cohesion -- table]]

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

\begin{itemize}%
\item [[analytic completion]]


\item [[localization of a ring]], [[localization of a commutative ring]]


\item [[completion of a module]]


\item [[formal scheme]]



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

A classical account is in section 1 of

\begin{itemize}%
\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}
Brief surveys include

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Completion_%28ring_theory%29}{Completion (ring theory)}}


\item Brett Bridges, \emph{An introduction to ring completions}, lecture notes 2011 (\href{http://www.math.uwo.ca/~srankin/courses/4123/2011/brett_presentation.pdf}{pdf})



\end{itemize}
Discussion of the [[derived functor]] of adic completion (``Greenless-May duality'') is in

\begin{itemize}%
\item [[John Greenlees]], [[Peter May]], \emph{Derived functors of I-adic completion and local homology}, J. Algebra 149 (1992), 438--453 (\href{http://math.uchicago.edu/~may/PAPERS/73.pdf}{pdf})


\item Leovigildo Alonso, Ana Jerem\'i{}as, [[Joseph Lipman]], \emph{Local Homology and Cohomology on Schemes} (\href{http://arxiv.org/abs/alg-geom/9503025}{arXiv:alg-geom/9503025})


\item Marco Porta, Liran Shaul, [[Amnon Yekutieli]], \emph{On the Homology of Completion and Torsion} (\href{http://arxiv.org/abs/1010.4386}{arXiv:1010.4386})



\end{itemize}
for more on this see also at \emph{\href{http://ncatlab.org/nlab/show/fracture+theorem#CompletionAndTorsionOnDerivedCategories}{fracture theorem -- Arithmetic fracturing for chain complexes}}

Discussion of interrelation between completion and [[etale morphisms]] is in

\begin{itemize}%
\item Leovigildo Alonso, Ana Jeremias, Marta Perez, \emph{Local structure theorems for smooth maps of formal schemes} (\href{http://arxiv.org/abs/math/0605115}{arXiv:math/0605115})

\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} (\href{http://arxiv.org/abs/0707.2585}{arXiv:0707.2585})

\end{itemize}
Completion for complex cobordism theory is in

\begin{itemize}%
\item [[John Greenlees]], [[Peter May]], \emph{Localization and Completion Theorems for MU-Module Spectra}, Annals of Mathematics Second Series, Vol. 146, No. 3 (Nov., 1997), pp. 509-544

\end{itemize}
[[!redirects completion of a ring]] [[!redirects completion of rings]] [[!redirects completions of a ring]] [[!redirects completions of rings]] [[!redirects completion (ring theory)]]

[[!redirects completion of a commutative ring]] [[!redirects completion of commutative rings]]

[[!redirects formal completion of a ring]] [[!redirects formal completion of rings]]



\end{document}