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. 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⁻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 an 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 sm/st = m/t for all s, t in S and m in M.

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)⁻¹M. 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. 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 an R-module M is trivial if and only if all its localizations at primes or maximal ideals are trivial.

Remarks

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 cannot 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. The corresponding S⁻¹R-module structure is given by ${\frac {a}{s}}\cdot {\frac {m}{t}}:={\frac {am}{st}}$ .

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 fact is foundational for the use of flatness in algebraic geometry, saying in particular that the inclusion of the open set Spec(S⁻¹R) 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.

References

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

Eisenbud, David (1995), Commutative algebra, Graduate Texts in Mathematics, 150, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94268-1, MR 1322960

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