# nLab completion of a module

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

Directly analogous to the concept of completion of a ring is the completion of a module over that ring.

In particular the formal completion or 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 ring spectra $Spec(A/\mathfrak{a}) \hookrightarrow Spec(A)$, so under the interpretation (see 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.

## Definition

### For modules over a commutative ring

###### Definition

For $A$ a commutative ring, $\mathfrak{a} \subset A$ an ideal in $A$ and for $N$ an $A$-module, then the $\mathfrak{a}$-adic completion or formal completion at $\mathfrak{a}$ of $N$ is the filtered limit

$N^{\wedge}_{\mathfrak{a}} \coloneqq \underset{\leftarrow}{\lim}_n N/(\mathfrak{a}^n N)$

of quotients of $N$ by the submodules induced by all powers of the ideal.

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

### For $\infty$-modules over an $E_\infty$-ring

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

#### Torsion approximation

###### Definition

An $A$-∞-module $N$ is an $\mathfrak{a}$-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 $a^k n = 0$.

###### Proposition
$A Mod_{\mathfrak{a}tor} \hookrightarrow A Mod$

is co-reflective and the co-reflector $\Pi_{\mathfrak{a}}$ – the torsion approximation – is smashing.

###### Proposition

For $N \in A Mod_{\leq 0}$ then torsion approximation, prop. , intuced a monomorphism on $\pi_0$

$\pi_0 \Pi_{\mathfrak{a}} N \hookrightarrow \pi_0 N$

including the $\mathfrak{a}$-nilpotent elements of $\pi_0 N$.

#### Localization

###### Definition

An $A$-∞-module $N$ is an $\mathfrak{a}$-local module if for every $\mathfrak{a}$-torsion module $T$ (def. ), the derived hom space

$Hom_A(T,N) \simeq \ast$

is contractible.

###### Proposition

For $\mathfrak{a} = (a)$ generated from a single element, then the localization of an (∞,1)-ring-map $A \to A[a^{-1}]$ is given by the (∞,1)-colimit over the sequence of right-multiplication with $a$

$A[a^{-1}] \simeq \underset{\rightarrow}{\lim} ( A \stackrel{\cdot a}{\longrightarrow} A \stackrel{\cdot a}{\longrightarrow} A \stackrel{\cdot a}{\longrightarrow} \cdots ) \,.$
###### Proposition
$A Mod_{\mathfrak{a}loc} \hookrightarrow A Mod$

of ∞-modules local away from $\mathfrak{a}$ is reflective. The reflector

$\Pi_{\mathfrak{a}dR} \colon A Mod \longrightarrow A Mod_{\mathfrak{a}loc}$

is called localization.

###### Proposition

There is a natural homotopy fiber sequence

$ʃ_{\mathfrak{a}} \longrightarrow id \longrightarrow ʃ_{\mathfrak{a}dR}$

relating $\mathfrak{a}$-torsion approximation on the left with $\mathfrak{a}$-localization on the right.

#### Completion

###### Definition/Proposition

An ∞-module $N$ over $A$ is $\mathfrak{a}$-complete if for all $\mathfrak{a}$-local $\infty$-modules $L$ (def. ) then $Hom_A(L,N) \simeq \ast$.

$A Mod_{\mathfrak{a}comp} \hookrightarrow A Mod$

of the (∞,1)-category of ∞-modules on the $\mathbb{a}$-complete ones is a reflective sub-(∞,1)-category. The reflector

$\flat_{\mathfrak{a}} \colon A Mod \longrightarrow A Mod_{\mathfrak{a}comp} \hookrightarrow A Mod$
$N \mapsto N^{\wedge}_{\mathfrak{a}}$

is called $\mathfrak{a}$-completion.

Definition relates to the traditional definition, def. , as follows

###### Proposition

Let $N$ a homotopically discrete ∞-module over the E-∞ ring $A$ which is a Noetherian module in that all its submodules are finitely finitely generated. Then the $\mathfrak{a}$-completion of $N$ in the sense of def. coincides with the traditional definition def. .

## Properties

### General properties

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

### 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 finitely generated ideal of its underlying commutative ring.

###### Proposition

The completion reflection $\flat_{\mathfrak{a}}$, def. , is a monoidal (∞,1)-functor.

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

###### Proposition

The full sub-(∞,1)-category of $\mathfrak{a}$-torsion modules, def. , is co-reflective

$A Mod_{\mathfrak{a}tor} \stackrel{\hookrightarrow}{\underset{\Pi_{\mathfrak{a}}}{\longleftarrow}} A Mod \,.$

Moreover, the coreflector $\Pi_{\mathfrak{a}}$ is “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

$\Omega (A[x_i^{-1}]/A) \longrightarrow A \longrightarrow A[x_i^{-1}] \,.$

From the general properties of smashing localization it follows that

###### Corollary

The coreflection $\Pi_{\mathfrak{a}} \colon A Mod \to A Mod$

1. preserves small (∞,1)-colimits;

2. is a “monoidal (∞,1)-functor” except possibly for preservation of units.

### Relation to localization

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

$(-)_{\mathfrak{a}}^{\wedge} \to id \to (-)[\mathfrak{a}^{-1}]$

with the completion functor of def. on the left and the localization functor of prop. on the right.

### Relation of formal completion to torsion approximation

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-modules forms together with $\mathfrak{a}$-torsion approximation an adjoint modality on the
(∞,1)-category of modules over $A$. See at fracture square for details.

cohesion modalitysymbolinterpretation
flat modality$\flat$formal completion at
shape modality$ʃ$torsion approximation
dR-shape modality$ʃ_{dR}$localization away
dR-flat modality$\flat_{dR}$adic residual
$\array{ && localization\;away\;from\;\mathfrak{a} && \stackrel{}{\longrightarrow} && \mathfrak{a}\;adic\;residual \\ & \nearrow & & \searrow & & \nearrow && \searrow \\ \Pi_{\mathfrak{a}dR} \flat_{\mathfrak{a}} X && && X && && \Pi_{\mathfrak{a}} \flat_{\mathfrak{a}dR} X \\ & \searrow & & \nearrow & & \searrow && \nearrow \\ && formal\;completion\;at\;\mathfrak{a}\; && \longrightarrow && \mathfrak{a}\;torsion\;approximation } \,,$

## References

Discussion in the context of higher algebra is in

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

• Gunnar Carlsson, Derived completions in stable homotopy theory, Journal of Pure and Applied Algebra Volume 212, Issue 3, March 2008, Pages 550–577 (arXiv:0707.2585)

Last revised on February 2, 2016 at 05:30:39. See the history of this page for a list of all contributions to it.