# 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 $R$ and a Gabriel filter $ℱ$ of left ideals in $R$, a Gabriel localization endofunctor

${G}_{ℱ}:{}_{R}\mathrm{Mod}\to {}_{R}\mathrm{Mod}$G_{\mathcal{F}} : {}_R Mod\to {}_R Mod

is defined in one of the number of equivalent ways.

For example, for any uniform filter $ℱ$ of left ideals in $R$ one defines a subfunctor of the identity functor ${\sigma }_{ℱ}$ on the category of left $R$-modules

$M↦{\sigma }_{ℱ}\left(M\right)={\sigma }_{ℒ}=\left\{m\in M\phantom{\rule{thinmathspace}{0ex}}\mid \phantom{\rule{thinmathspace}{0ex}}m\in M,\exists m\in M,\phantom{\rule{thinmathspace}{0ex}}\mathrm{Jm}=0\right\}\subset M$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 ${\sigma }_{ℒ}$ was called a radical functor. If $ℱ$ is not only uniform but in fact a Gabriel filter then the radical ${\sigma }_{ℱ}$ is idempotent, i.e. ${\sigma }_{ℱ}^{2}={\sigma }_{ℱ}$. If $R$ is unital, ${\sigma }_{ℱ}$ is equivalent to the functor given on objects by

$\sigma {\prime }_{ℒ}\left(M\right)={\mathrm{colim}}_{J\in ℱ}{\mathrm{Hom}}_{R}\left(R/J,M\right)$\sigma'_{\mathcal{L}}(M) = colim_{J\in\mathcal{F}} Hom_R(R/J,M)

For each uniform fiter $ℱ$ one also defines the endofunctor ${H}_{ℱ}$ on ${}_{R}\mathrm{Mod}$ by

${H}_{ℱ}\left(M\right)={\mathrm{colim}}_{J\in ℱ}{\mathrm{Hom}}_{R}\left(J,M\right)$H_{\mathcal{F}}(M) = colim_{J\in\mathcal{F}} Hom_R(J,M)

(the colimit is over downward directed family of ;eft ideals in $ℱ$ 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 $R$-module on the colimit group ${H}_{ℱ}\left(M\right)$).

Finally, for the Gabriel filter $ℱ$ one defines the Gabriel (endo)functor ${G}_{ℱ}$ on objects by

${G}_{ℱ}\left(M\right):={H}_{ℱ}\left(M/{\sigma }_{ℱ}\left(M\right)\right)={\mathrm{colim}}_{J\in ℒ}{\mathrm{Hom}}_{R}\left(J,M/{\sigma }_{ℱ}\left(M\right)\right)$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 ${G}_{ℱ}$ is the localized category. The left $R$-module ${G}_{ℱ}\left(R\right)$ has a canonical structure of a ring over $R$; there is a natural forgetful functor from the localized category to the category of left ${G}_{ℱ}\left(R\right)$-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)