nLab
Gabriel localization

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

• Pierre Gabriel, Des Categories Abeliennes

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

$$M\mapsto {\sigma }_{ℱ}(M)={\sigma }_{ℒ}=\{m\in M\mid m\in M,\exists m\in M,\mathrm{Jm}=0\}\subset M$$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

$$\sigma {\prime }_{ℒ}(M)={\mathrm{colim}}_{J\in ℱ}{\mathrm{Hom}}_R(R/J,M)$$\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

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

(the colimit is over downward directed family of ;eft ideals in $ℱ$ 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 $H_{ℱ}(M)$).

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

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