nLab localization of a module

Idea

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

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

Consider a reflective localization functor

$$Q^* = Q^*_\Sigma \colon \mathcal{A}\to \Sigma^{-1}\mathcal{A}$$

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

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

Depending on an author or a context, we say that a (reflective) localization functor of category of modules is flat if either $Q^*$ is also left exact functor, or more strongly that the composed endofunctor $Q_* Q^*$ 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 ($\mathbb{S}$-modules)

Claim

Suppose that $L \colon Spectra \to Spectra$ is a smashing localization given by smash product with some spectrum $T$. Write $F$ for the homotopy fiber

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

Then there is a fracture diagram of operations

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

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

(Charles Rezk, MO comment,August 2014)

Example

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

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

and hence

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

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

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

• $[T,-]$ is forming $p$-adic residual.

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

With (Bousfield 79, prop.2.5)

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

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

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.

Properties

Eilenberg-Watts theorem

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

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

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

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

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

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

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

Relation with torsion approximation and completion

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:

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

Related concepts

• localization of an abelian group

• localization of a ring, Ore localization, Gabriel localization, Cohn localization

• localization of a space (and of a spectrum)

• localization of a category (= localization functor)

• locally free module

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. ?koda?, 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

• Aldridge Bousfield, Daniel Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol 304, Springer 1972

• Aldridge Bousfield, The localization of spectra with respect to homology , Topology vol 18 (1979) (pdf)

• Dennis Sullivan, 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 (pdf)

Discussion in the context of higher algebra is in

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