# Localization of a module

In algebraic geometry, the **localization of a module** is a construction to introduce denominators in a module for a ring. More precisely, it is a systematic way to construct a new module *S*^{−1}*M* out of a given module *M* containing algebraic fractions

where the denominators *s* range in a given subset *S* of *R*.

The technique has become fundamental, particularly in algebraic geometry, as the link between modules and sheaf theory. Localization of a module generalizes localization of a ring.

## Definition

In this article, let *R* be a commutative ring and *M* an *R*-module.

Let *S* a multiplicatively closed subset of *R*, i.e. 1 ∈ *S* and for any *s* and *t* ∈ *S*, the product *st* is also in *S*. Then the **localization of M with respect to S**, denoted

*S*

^{−1}

*M*, is defined to be the following module: as a set, it consists of equivalence classes of pairs (

*m*,

*s*), where

*m*∈

*M*and

*s*∈

*S*. Two such pairs (

*m*,

*s*) and (

*n*,

*t*) are considered equivalent if there is a third element

*u*of

*S*such that

*u*(*sn*-*tm*) = 0

It is common to denote these equivalence classes

To make this set a *R*-module, define

and

(*a* ∈ *R*). It is straightforward to check that the definition is well-defined, i.e. yields the same result for different choices of representatives of fractions. One interesting characterization of the equivalence relation is that it is the smallest relation (considered as a set) such that cancellation laws hold for elements in *S*. That is, it is the smallest relation such that *rs/us = r/u* for all *s* in *S*.

One case is particularly important: if *S* equals the complement of a prime ideal *p* ⊂ *R* (which is multiplicatively closed by definition of prime ideals) then the localization is denoted *M*_{p} instead of (*R*\*p*)^{−1}*M*. The **support of the module** *M* is the set of prime ideals *p* such that *M*_{p} ≠ 0. Viewing *M* as a function from the spectrum of *R* to *R*-modules, mapping

this corresponds to the support of a function.
Localization of a module at primes also reflects the "local properties" of the module. In particular, there are many cases where the more general situation can be reduced to a statement about localized modules. The reduction is because a *R*-module *M* is trivial if and only if all its localizations at primes or maximal ideals are trivial.

## Remarks

- The definition applies in particular to
*M*=*R*, and we get back the localized ring*S*^{−1}*R*.

- There is a module homomorphism

- φ:
*M*→*S*^{−1}*M*

- φ:
- mapping
- φ(
*m*) =*m*/ 1.

- φ(
- Here φ need not be injective, in general, because there may be significant torsion. The additional
*u*showing up in the definition of the above equivalence relation can not be dropped (otherwise the relation would not be transitive), unless the module is torsion-free.

- Some authors allow not necessarily multiplicatively closed sets
*S*and define localizations in this situation, too. However, saturating such a set, i.e. adding*1*and finite products of all elements, this comes down to the above definition.

## Tensor product interpretation

By the very definitions, the localization of the module is tightly linked to the one of the ring via the tensor product

*S*^{−1}*M*=*M*⊗_{R}*S*^{−1}*R*,

This way of thinking about localising is often referred to as extension of scalars.

As a tensor product, the localization satisfies the usual universal property.

## Flatness

From the definition, one can see that localization of modules is an exact functor, or in other words (reading this in the tensor product) that *S*^{−1}*R* is a flat module over *R*. This fact is foundational for the use of flatness in algebraic geometry, saying in particular that the inclusion of the open set Spec(*S*^{−1}*R*) into Spec(*R*) (see spectrum of a ring) is a flat morphism.

## (Quasi-)coherent sheaves

In terms of localization of modules, one can define quasi-coherent sheaves and coherent sheaves on locally ringed spaces. In algebraic geometry, the **quasi-coherent** *O*_{X}-**modules** for schemes *X* are those that are locally modelled on sheaves on Spec(*R*) of localizations of any *R*-module *M*. A **coherent** *O*_{X}-**module** is such a sheaf, locally modelled on a finitely-presented module over *R*.

## See also

### Localization

Category:Localization (mathematics)

- Local analysis
- Localization (algebra)
- Localization of a category
- Localization of a ring
- Localization of a topological space

## References

Any textbook on commutative algebra covers this topic, such as:

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