Localization of A Module - Definition

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

.

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)−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

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.