Localization of a module

Suppose R is a commutative ring, and M is a given R-module. The construction of the localization of a ring is a systematic way to provide multiplicative inverse elements for any given subset S of R; it is no loss of generality to assume S is closed under multiplication, and contains 1. The localized ring is written S^-1R, in one possible notation; there is a ring homomorphism

φ:R→S^-1R

φ(r) = r/1.

To construct a corresponding S^-1R-module S^-1M will involve making the action of any s in S invertible on the localized module S^-1M, when it need not be on M. Therefore while there should be a module homomorphism

ψ:M → S^-1M

It is not hard to supply this, on the basis of general theory (for example, adjoint functors). It turns out that in this case there are two quite natural constructions, each of which has some advantages. Firstly, we can define S^-1M simply by extension of scalars, as

S^-1M = M⊗_RS^-1R,

m/s in M×S

One immediate use for the 'fractions' approach is to show that localization of modules is an exact functor, or in other words (reading this in the first construction) that S^-1R is a flat module over R. This is actually foundational for the use of flatness in algebraic geometry, saying that the inclusion of an open set in Spec(R) (see spectrum of a ring) is a flat morphism.

In terms of localization of modules, one can define the ideas of quasi-coherent sheaf and coherent sheaf, for locally ringed spaces. In algebraic geometry, the quasi-coherent O_X-modules for schemess 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.