Let be a commutative ring with unit and let be a multiplicative submonoid of . Localization at is a functor
Writing for the canonical ring localization homomorphism, we can use it to put an -module structure on any -module in a functorial way. In other words, we have a functor (restriction of scalars) Since is isomorphic to the full subcategory of consisting of modules in which the action of every is invertible, we can think of as a forgetful functor!
Claim. Localization is left adjoint to restriction of scalars: . Spelled out, this means there is a natural isomorphism of sets
Thinking of as a forgetful functor, this shines new light on localization of modules as a sort of “best inverse” to forgetting that one can multiply by -fractions, or more precisely some sort of free construction (recall that free constructions are left adjoints to forgetful functors). In other words, and as expected, the localization is the “smallest” or “free-est” module in which one can divide by elements of .
Proof of the claim. Recall that we characterized localization via universal arrows: for any -linear map , there exists a unique -linear map such that , where is the universal arrow.
We define by sending some -linear map to . The previous paragraph tells us that is a bijection.
The last thing to check is naturality. It suffices to check that for any -linear map , we have . However, this is immediate because we defined as the unique -linear map such that this is verified!
Now recall that extension of scalar has a more “usual” left adjoint, called extension of scalars, denoted in our case by Its action on objects is given by . Because of the unicity of adjoints up to a unique canonical isomorphism, there is a natural isomorphism of functors So localization enjoys the same universal property as this tensor product, and similarily this tensor product enjoys the same universal property as the localization.