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 $R$ a (possibly noncommutative) unital ring, let $𝒜=R$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

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

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

Properties

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

$$Q^{*}(M)=Q^{*}(R){\otimes }_{R}M$$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

$$Q^{*}(M)=S^{-1}R{\otimes }_{R}M$$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 $S^{-1}M$.

Related concepts

• localization of a ring, Ore localization, Gabriel localization, Cohn localization
• localization of a category (= localization functor)
• locally free module

Literature

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