Localization of a module
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In mathematics, the localization of a module is a construction to introduce denominators in a module M for a ring R. It has become fundamental in particular in algebraic geometry, as the link between modules and sheaf theory.
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
that is thought of in fractional terms as
- φ(r) = r/1.
Here φ need not be injective, in general.
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 will not generally be injective, and 'collapsing' can be expected in some cases, because there may be significant torsion. What is clearly required is some universal property statement that defines the solution.
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,
which reduces the construction to general facts about tensor products. Or we may imitate the construction of fractions, by specifying how symbols
- m/s in M×S
are to be handled. Appealing to general results on universal properties, one sees that these two constructions are equivalent.
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 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.