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⁻¹M out of a given module M containing algebraic fractions

{\frac {m}{s}}

.

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. for any s and t ∈ S, the product st is also in S. Then the localization of M with respect to S, denoted S⁻¹M, 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

{\frac {m}{s}}

.

To make this set a R-module, define

{\frac {m}{s}}+{\frac {n}{t}}:={\frac {tm+sn}{st}}

and

a\cdot {\frac {m}{s}}:={\frac {am}{s}}

(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.

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)⁻¹M.

Local property

One says that a property P for a R-module M (or morphisms between modules) is a local property if the following are equivalent.

(i) P holds for M.
(ii) P holds for $M_{\mathfrak {p}}$ for all prime ideal ${\mathfrak {p}}$ of R.
(iii) P holds for $M_{\mathfrak {m}}$ for all maximal ideal ${\mathfrak {m}}$ of R.

In other words, to show a local property holds for M, it is enough to show it holds for M at every maximal ideal.

For example, $M=0$ if and only if $M_{\mathfrak {m}}=0$ for all maximal ideal ${\mathfrak {m}}$ of R. (This is a special case of the notion of support; for this, see below.) In practice, this is often used in the following way: to show $M=N$ , it is enough to $M_{\mathfrak {m}}=N_{\mathfrak {m}}$

Let $M,N$ be R-modules. Then the following are local properties.

M is flat module.
$f:M\to N$ is injective.
$f:M\to N$ is surjective.

On the other hand, some properties are not local properties. For example, "noetherian" is (in general) not a local property: that is, to say there is a non-noetherian ring whose localization at every maximal ideal is noetherian: this example is due to Nagata.^{[citation needed]}

Support

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

p\mapsto M_{p}

this corresponds to the support of a function.

Remarks

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

There is a module homomorphism

φ: M → S⁻¹M

mapping

φ(m) = m / 1.

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⁻¹M = M ⊗_RS⁻¹R,

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⁻¹R is a flat module over R. This is actually foundational for the use of flatness in algebraic geometry, saying in particular that the inclusion of an open set in 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.

References

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

Eisenbud, David (1995), Commutative algebra, Graduate Texts in Mathematics, vol. 150, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94268-1; 978-0-387-94269-8, MR1322960 {{citation}}: Check |isbn= value: invalid character (help)