nLab
Gabriel localization

Pierre Gabriel introduced a number of constructions in localization theory, mostly in abelian context in his thesis published as

and in general context in his book with Zisman. By Gabriel localization one usually means a specific class of localizations of rings and the corresponding localization of categories of modules over rings.

Given a (possibly noncommutative and nonunital) ring RR and a Gabriel filter \mathcal{F} of left ideals in RR, a Gabriel localization endofunctor

G : RMod RMod G_{\mathcal{F}} : {}_R Mod\to {}_R Mod

is defined in one of the number of equivalent ways.

For example, for any uniform filter \mathcal{F} of left ideals in RR one defines a subfunctor of the identity functor σ \sigma_{\mathcal{F}} on the category of left RR-modules

Mσ (M)=σ ={mM|mM,mM,Jm=0}M M\mapsto \sigma_{\mathcal{F}}(M) = \sigma_{\mathcal{L}} = \{m\in M \,|\, m\in M, \exists m\in M,\, Jm = 0\}\subset M

In a later work of Goldman σ \sigma_{\mathcal{L}} was called a radical functor. If \mathcal{F} is not only uniform but in fact a Gabriel filter then the radical σ \sigma_{\mathcal{F}} is idempotent, i.e. σ 2=σ \sigma_{\mathcal{F}}^2 = \sigma_{\mathcal{F}}. If RR is unital, σ \sigma_{\mathcal{F}} is equivalent to the functor given on objects by

σ (M)=colim JHom R(R/J,M) \sigma'_{\mathcal{L}}(M) = colim_{J\in\mathcal{F}} Hom_R(R/J,M)

For each uniform fiter \mathcal{F} one also defines the endofunctor H H_{\mathcal{F}} on RMod{}_R Mod by

H (M)=colim JHom R(J,M) H_{\mathcal{F}}(M) = colim_{J\in\mathcal{F}} Hom_R(J,M)

(the colimit is over downward directed family of ;eft ideals in \mathcal{F} and is a colimit of a functor with values in the category of abelian groups; the uniformness condition however gurantees that there is a canonical structure of an RR-module on the colimit group H (M)H_{\mathcal{F}}(M)).

Finally, for the Gabriel filter \mathcal{F} one defines the Gabriel (endo)functor G G_{\mathcal{F}} on objects by

G (M):=H (M/σ (M))=colim JHom R(J,M/σ (M)) G_{\mathcal{F}}(M) := H_{\mathcal{F}}(M/\sigma_{\mathcal{F}}(M)) = colim_{J\in\mathcal{L}}Hom_R(J,M/\sigma_{\mathcal{F}}(M))

The essential image of the functor G G_{\mathcal{F}} is the localized category. The left RR-module G (R)G_{\mathcal{F}}(R) has a canonical structure of a ring over RR; there is a natural forgetful functor from the localized category to the category of left G (R)G_{\mathcal{F}}(R)-modules. Under strong assumptions on the filter this functor is in fact an equivalence of categories, e.g. when the localization is Ore.

  • Zoran Škoda, Localizations for construction of quantum coset spaces, in “Noncommutative geometry and Quantum groups”, W.Pusz, P.M. Hajac, eds. Banach Center Publications vol.61, pp. 265–298, Warszawa 2003, math.QA/0301090.
  • Zoran Škoda, Noncommutative localization in noncommutative geometry, London Math. Society Lecture Note Series 330, ed. A. Ranicki; pp. 220–313, [math.QA/0403276], (http://arxiv.org/abs/math.QA/0403276)
Revised on June 8, 2011 17:07:22 by Zoran Škoda (161.53.130.104)