nLab
localization of a module

Contents

Idea

A localization of a module is the result of application of an additive localization functor on a category of modules over some ring RR.

When RR is a commutative ring of functions, and under the interpretation of modules as generalized vector bundles the localization of a module corresponds to the restriction of the bundle to a subspace of its base space.

Definition

For RR a (possibly noncommutative) unital ring, let 𝒜=R\mathcal{A} = RMod be the category RR-modules. Here RR may be the structure sheaf of some ringed topos and accordingly the modules may be sheaves of modules.

Consider a reflective localization functor

Q *=Q Σ *:𝒜Σ 1𝒜 Q^* = Q^*_\Sigma \colon \mathcal{A}\to \Sigma^{-1}\mathcal{A}

with right adjoint Q *Q_*. The application of this functor to a module M𝒜M\in \mathcal{A} is some object Q *(M)Q^*(M) in the localized category Σ 1𝒜\Sigma^{-1}\mathcal{A}, which is up to isomorphism determined by its image Q *Q *(M)Q_* Q^*(M).

The localization map is the component of the unit of the adjunction (usually denoted by ii, jj or ι\iota in this setup) ι M:MQ *Q *(M)\iota_M : M\to Q_* Q^*(M).

Depending on an author or a context, we say that a (reflective) localization functor of category of modules is flat if either Q *Q^* is also left exact functor, or more strongly that the composed endofunctor Q *Q *Q_* Q^* is exact. For example, Gabriel localization is flat in the first, weak sense, and left or right Ore localization is flat in the second, stronger, sense.

Properties

By the Eilenberg-Watts theorem, if 𝒜=R\mathcal{A}= RMod then the localization of a module

Q *(M)=Q *(R) RM Q^*(M) = Q^*(R)\otimes_R M

is given by forming the tensor product of modules with the localizatin of the ring RR, regarded as a module over itself.

If the localization is a left Ore localization or commutative localization at a set SRS\subset R then Q *(R)=S 1RQ^*(R) = S^{-1} R is the localization of the ring itself and hence in this case the localization of the module

Q *(M)=S 1R RM Q^*(M) = S^{-1} R\otimes_R M

is given by extension of scalars along the localization map RS 1RR \to S^{-1}R of the ring itself.

In these cases there are also direct constructions of Q *(M)Q^*(M) (not using to Q *(R)Q^*(R)) which give an isomorphic result, also denoted by S 1MS^{-1}M.

Literature

Standard discussion over commutative rings is for instance in

  • Andreas Gathmann, Localization (pdf)

Discussion in the general case of noncommutative geometry is in

  • Z. Škoda, Noncommutative localization in noncommutative geometry, London Math. Society Lecture Note Series 330 (pdf), ed. A. Ranicki; pp. 220–313, math.QA/0403276

Revised on July 24, 2014 09:47:23 by Urs Schreiber (89.204.155.107)