nLab completion of a module

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 $a^k n = 0$.

(Lurie “Completions”, def. 4.1.3).

Proposition

The full sub-(∞,1)-category

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

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

(Lurie “Completions”, prop. 4.1.12).

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 $\pi_0 N$.

(Lurie “Completions”, prop. 4.1.18).

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.

(Lurie “Completions”, def. 4.1.9).

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 $a$

$$A[a^{-1}] \simeq \underset{\rightarrow}{\lim} ( A \stackrel{\cdot a}{\longrightarrow} A \stackrel{\cdot a}{\longrightarrow} A \stackrel{\cdot a}{\longrightarrow} \cdots ) \,.$$

(Lurie “Completions”, remark 4.1.11)

Proposition

The full sub-(∞,1)-category

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

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 $Hom_A(L,N) \simeq \ast$.

The full sub-(∞,1)-category

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

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.

(Lurie “Completions”, def. 4.2.1, lemma 4.2.2).

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. .

(Lurie “Completions”, prop. 4.3.6)

Properties

General properties

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

(Lurie “Completions”, prop. 4.1.17)

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.

(Lurie “Completions”, remark 4.2.6).

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

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

(Lurie “Completions”, prop. 4.1.12).

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.

See also (Lurie “Completions”, cor. 4.1.16).

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.

(Lurie “Completions”, example 4.1.14, remark 4.1.20)

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 in E-∞ arithmetic geometry:

cohesion modality	symbol	interpretation
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

the differential cohomology hexagon/arithmetic fracture squares:

$$\array{ && localization\;away\;from\;\mathfrak{a} && \stackrel{}{\longrightarrow} && \mathfrak{a}\;adic\;residual \\ & \nearrow & & \searrow & & \nearrow && \searrow \\ \Pi_{\mathfrak{a}dR} \flat_{\mathfrak{a}} X && && X && && \Pi_{\mathfrak{a}} \flat_{\mathfrak{a}dR} X \\ & \searrow & & \nearrow & & \searrow && \nearrow \\ && formal\;completion\;at\;\mathfrak{a}\; && \longrightarrow && \mathfrak{a}\;torsion\;approximation } \,,$$

References

Discussion in the context of higher algebra is in

• Jacob Lurie, section 4 of Proper Morphisms, Completions, and the Grothendieck Existence Theorem

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)