Localization of a module

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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[edit]

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 tS, 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 mM and sS. 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{a m}{s}

(aR). 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 pR (which is multiplicatively closed by definition of prime ideals) then the localization is denoted Mp instead of (R\p)−1M. The support of the module M is the set of prime ideals p such that Mp ≠ 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 a R-module M is trivial if and only if all its localizations at primes or maximal ideals are trivial.

Remarks[edit]

  • The definition applies in particular to M=R, and we get back the localized ring S−1R.
  • There is a module homomorphism
φ: MS−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[edit]

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

S−1M = MRS−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[edit]

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[edit]

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 OX-modules for schemes X are those that are locally modelled on sheaves on Spec(R) of localizations of any R-module M. A coherent OX-module is such a sheaf, locally modelled on a finitely-presented module over R.

See also[edit]

Localization[edit]

Category:Localization (mathematics)

References[edit]

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