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
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
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
For each uniform fiter $\mathcal{F}$ one also defines the endofunctor $H_{\mathcal{F}}$ on ${}_R Mod$ by
(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
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.
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
Last revised on August 20, 2024 at 22:52:12. See the history of this page for a list of all contributions to it.