completion of a module




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Formal geometry



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 AA at an ideal 𝔞\mathfrak{a} has a corresponding analog for modules. Where the adic completion A 𝔞 A_{\mathfrak{a}}^\wedge of the ring itself has the geometric interpretation of forming the formal neighbourhood Spf(A 𝔞 )Spf(A_{\mathfrak{a}}^\wedge) of ring spectra Spec(A/𝔞)Spec(A)Spec(A/\mathfrak{a}) \hookrightarrow Spec(A), so under the interpretation (see here) of AA-modules as bundles over Spec(A)Spec(A), the 𝔞\mathfrak{a}-adic completion N 𝔞 N_{\mathfrak{a}}^\wedge of an AA-module NN has the interpretation of being the restriction of that bundle to that formal neighbourhood.


For modules over a commutative ring


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

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

of quotients of NN by the submodules induced by all powers of the ideal.

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

For \infty-modules over an E E_\infty-ring

Let AA be an E-∞ ring and 𝔞π 0A\mathfrak{a} \subset \pi_0 A a finitely generated ideal of its underlying commutative ring.

Torsion approximation


An AA-∞-module NN is an 𝔞\mathfrak{a}-torsion module if for all elements nπ kNn \in \pi_k N and all elements a𝔞a \in \mathfrak{a} there is kk \in \mathbb{N} such that a kn=0a^k n = 0.

(Lurie “Completions”, def. 4.1.3).


The full sub-(∞,1)-category

AMod 𝔞torAMod A Mod_{\mathfrak{a}tor} \hookrightarrow A Mod

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

(Lurie “Completions”, prop. 4.1.12).


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

π 0Π 𝔞Nπ 0N \pi_0 \Pi_{\mathfrak{a}} N \hookrightarrow \pi_0 N

including the 𝔞\mathfrak{a}-nilpotent elements of π 0N\pi_0 N.

(Lurie “Completions”, prop. 4.1.18).



An AA-∞-module NN is an 𝔞\mathfrak{a}-local module if for every 𝔞\mathfrak{a}-torsion module TT (def. ), the derived hom space

Hom A(T,N)* Hom_A(T,N) \simeq \ast

is contractible.

(Lurie “Completions”, def. 4.1.9).


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

A[a 1]lim(AaAaAa). 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)


The full sub-(∞,1)-category

AMod 𝔞locAMod A Mod_{\mathfrak{a}loc} \hookrightarrow A Mod

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

Π 𝔞dR:AModAMod 𝔞loc \Pi_{\mathfrak{a}dR} \colon A Mod \longrightarrow A Mod_{\mathfrak{a}loc}

is called localization.


There is a natural homotopy fiber sequence

ʃ 𝔞idʃ 𝔞dR ʃ_{\mathfrak{a}} \longrightarrow id \longrightarrow ʃ_{\mathfrak{a}dR}

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



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

The full sub-(∞,1)-category

AMod 𝔞compAMod A Mod_{\mathfrak{a}comp} \hookrightarrow A Mod

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

𝔞:AModAMod 𝔞compAMod \flat_{\mathfrak{a}} \colon A Mod \longrightarrow A Mod_{\mathfrak{a}comp} \hookrightarrow A Mod
NN 𝔞 N \mapsto N^{\wedge}_{\mathfrak{a}}

is called 𝔞\mathfrak{a}-completion.

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

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


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

(Lurie “Completions”, prop. 4.3.6)


General properties

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

(Lurie “Completions”, prop. 4.1.17)


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

Let AA be an E-∞ ring and 𝔞π 0A\mathfrak{a} \subset \pi_0 A a finitely generated ideal of its underlying commutative ring.


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

(Lurie “Completions”, remark 4.2.6).

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


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

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

Moreover, the coreflector Π 𝔞\Pi_{\mathfrak{a}} is “smashing”, in that there is VAModV \in A Mod such that Π 𝔞()V()\Pi_{\mathfrak{a}}(-) \simeq V \wedge (-) is given by the smash product with VV. If 𝔞=({x i} i)\mathfrak{a} = (\{x_i\}_i) then VV is the tensor product V=iV iV =\underset{i}{\otimes} V_i over all the homotopy fibers

Ω(A[x i 1]/A)AA[x i 1]. \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


The coreflection Π 𝔞:AModAMod\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 𝔞\mathfrak{a}-completion Π 𝔞\Pi_{\mathfrak{a}} is localization away from 𝔞\mathfrak{a}, in that there is a homotopy fiber sequence

() 𝔞 id()[𝔞 1] (-)_{\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 𝔞A\mathfrak{a}\subset A of a commutative ring AA or more generally of an E-∞ ring, then the derived functor of 𝔞\mathfrak{a}-adic completion of A-modules forms together with 𝔞\mathfrak{a}-torsion approximation an adjoint modality on the
(∞,1)-category of modules over AA. See at fracture square for details.

cohesion in E-∞ arithmetic geometry:

cohesion modalitysymbolinterpretation
flat modality\flatformal completion at
shape modalityʃʃtorsion approximation
dR-shape modalityʃ dRʃ_{dR}localization away
dR-flat modality dR\flat_{dR}adic residual

the differential cohomology hexagon/arithmetic fracture squares:

localizationawayfrom𝔞 𝔞adicresidual Π 𝔞dR 𝔞X X Π 𝔞 𝔞dRX formalcompletionat𝔞 𝔞torsionapproximation, \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 } \,,


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.