nLab Gabriel localization

Idea

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 $\mathcal{F}$ 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

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

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

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

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 left 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.

Related entries

• Calculus of fractions and homotopy theory

References

• Pierre Gabriel, Des Categories Abeliennes, Bulletin de la Société Mathématique de France 90 (1962) 323-448 [numdam:BSMF_1962__90__323_0]

• Pierre Gabriel, La localisation dans les anneaux non commutatifs, Séminaire Dubreil (1959-1960) exposé 2, 1-35 [numdam:SD_1959-1960__13_1_A2_0, pdf]

• Alexander L. Rosenberg, Non-commutative affine semischemes and schemes, Seminar on supermanifolds 26, Dept. Math., U. Stockholm (1988) pdf

• 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