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 $\mathcal{F}$ of left ideals in $R$, a Gabriel localization endofunctor

$G_{\mathcal{F}} : {}_R Mod\to {}_R Mod$

is defined in one of the number of equivalent ways.

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

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

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

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

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

$H_{\mathcal{F}}(M) = colim_{J\in\mathcal{F}} Hom_R(J,M)$

(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 $R$-module on the colimit group $H_{\mathcal{F}}(M)$).

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

$G_{\mathcal{F}}(M) := H_{\mathcal{F}}(M/\sigma_{\mathcal{F}}(M))
= colim_{J\in\mathcal{L}}Hom_R(J,M/\sigma_{\mathcal{F}}(M))$

The essential image of the functor $G_{\mathcal{F}}$ is the localized category. The left $R$-module $G_{\mathcal{F}}(R)$ has a canonical structure of a ring over $R$; there is a natural forgetful functor from the localized category to the category of left $G_{\mathcal{F}}(R)$-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

Revised on November 19, 2014 07:10:45
by Tim Porter
(2.27.158.74)