nLab
localization of a module

Contents

Idea

A localization of a module is the result of application of an additive localization functor on a category of modules over some ring RR.

When RR is a commutative ring of functions, and under the interpretation of modules as generalized vector bundles the localization of a module corresponds to the restriction of the bundle to a subspace of its base space.

Definition

For modules over rings

For RR a (possibly noncommutative) unital ring, let 𝒜=R\mathcal{A} = RMod be the category RR-modules. Here RR may be the structure sheaf of some ringed topos and accordingly the modules may be sheaves of modules.

Consider a reflective localization functor

Q *=Q Σ *:𝒜Σ 1𝒜 Q^* = Q^*_\Sigma \colon \mathcal{A}\to \Sigma^{-1}\mathcal{A}

with right adjoint Q *Q_*. The application of this functor to a module M𝒜M\in \mathcal{A} is some object Q *(M)Q^*(M) in the localized category Σ 1𝒜\Sigma^{-1}\mathcal{A}, which is up to isomorphism determined by its image Q *Q *(M)Q_* Q^*(M).

The localization map is the component of the unit of the adjunction (usually denoted by ii, jj or ι\iota in this setup) ι M:MQ *Q *(M)\iota_M : M\to Q_* Q^*(M).

Depending on an author or a context, we say that a (reflective) localization functor of category of modules is flat if either Q *Q^* is also left exact functor, or more strongly that the composed endofunctor Q *Q *Q_* Q^* is exact. For example, Gabriel localization is flat in the first, weak sense, and left or right Ore localization is flat in the second, stronger, sense.

For chain complexes

… Greenlees-May duality…

For spectra (𝕊\mathbb{S}-modules)

Claim

Suppose that L:SpectraSpectraL \colon Spectra \to Spectra is a smashing localization given by smash product with some spectrum TT. Write FF for the homotopy fiber

F𝕊T. F \longrightarrow \mathbb{S} \longrightarrow T \,.

Then there is a fracture diagram of operations

T() [T,] 𝕊 [F,] F() \array{ T \wedge (-) && \longleftarrow && [T,-] \\ & \nwarrow && \swarrow \\ && \mathbb{S} \\ & \swarrow & & \nwarrow \\ [F,-] && \longleftarrow && F \wedge (-) }

where [F,][F,-] and T():SpectrSpectraT \wedge (-) \colon Spectr \to Spectra are idempotent (∞,1)-monads and [T,][T,-], [F,][F,-] are idempotent \infty-comonad, the diagonals are homotopy fiber sequences.

(Charles Rezk, MO comment,August 2014)

Example

For T=S[p 1]T = S \mathbb{Z}[p^{-1}] the Moore spectrum of the integers localized away from pp, then

F=Σ 1S([p 1]/)𝕊S[p 1] F = \Sigma^{-1} S (\mathbb{Z}[p^{-1}]/\mathbb{Z}) \to \mathbb{S} \to S \mathbb{Z}[p^{-1}]

and hence

  • Σ 1S([p 1]/)()\Sigma^{-1} S (\mathbb{Z}[p^{-1}]/\mathbb{Z}) \wedge (-) is pp-torsion approximation;

  • [Σ 1S([p 1]/),][\Sigma^{-1} S (\mathbb{Z}[p^{-1}]/\mathbb{Z}),-] is pp-completion;

  • S[p 1]()S \mathbb{Z}[p^{-1}] \wedge (-) is localization away from pp (pp-rationalization)

  • [T,][T,-] is forming pp-adic residual.

localizationawayfrom𝔞 𝔞adicresidual X formalcompletionat𝔞 𝔞torsionapproximation, \array{ && localization\;away\;from\;\mathfrak{a} && \stackrel{}{\longrightarrow} && \mathfrak{a}\;adic\;residual \\ & \nearrow & & \searrow & & \nearrow && \searrow \\ && && X && && \\ & \searrow & & \nearrow & & \searrow && \nearrow \\ && formal\;completion\;at\;\mathfrak{a}\; && \longrightarrow && \mathfrak{a}\;torsion\;approximation } \,,

With (Bousfield 79, prop.2.5)

For \infty-modules over E E_\infty-rings

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

Definition

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

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

is contractible.

(Lurie “Completions”, def. 4.1.9).

Proposition

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)

Proposition

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.

Proposition

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.

Properties

Eilenberg-Watts theorem

By the Eilenberg-Watts theorem, if 𝒜=R\mathcal{A}= RMod then the localization of a module

Q *(M)=Q *(R) RM Q^*(M) = Q^*(R)\otimes_R M

is given by forming the tensor product of modules with the localizatin of the ring RR, regarded as a module over itself.

If the localization is a left Ore localization or commutative localization at a set SRS\subset R then Q *(R)=S 1RQ^*(R) = S^{-1} R is the localization of the ring itself and hence in this case the localization of the module

Q *(M)=S 1R RM Q^*(M) = S^{-1} R\otimes_R M

is given by extension of scalars along the localization map RS 1RR \to S^{-1}R of the ring itself.

In these cases there are also direct constructions of Q *(M)Q^*(M) (not using to Q *(R)Q^*(R)) which give an isomorphic result, also denoted by S 1MS^{-1}M.

Relation with torsion approximation and completion

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 } \,,

Literature

Standard discussion over commutative rings is for instance in

  • Andreas Gathmann, Localization (pdf)

Discussion in the general case of noncommutative geometry is in

  • Z. Škoda, Noncommutative localization in noncommutative geometry, London Math. Society Lecture Note Series 330 (pdf), ed. A. Ranicki; pp. 220–313, math.QA/0403276

Discussion in the context of spectra originates in

Discussion in the context of higher algebra is in

Revised on August 19, 2014 01:32:25 by Urs Schreiber (193.175.4.212)