nLab Gabriel filter

Definition

A Gabriel filter $G$ is a uniform filter of left ideals in a ring $R$ which is idempotent under Gabriel composition of filters.

Terminology

In our definition this notion is equivalent to topologizing filter; though for some authors the latter notion slightly differs. Stenstroem says Gabriel topology instead of Gabriel filter, because all Gabriel filters form a basis of neighborhoods of $0$ for a topology on $R$; on the other hand the Gabriel filter itself is an additive analogue of Grothendieck topology, see also enriched sheaf.

Properties

If $L$ and $L'$ are left ideals in a Gabriel filter $F$, then the set $L L'$ (of all products $l l'$ where $l\in L, l'\in L'$) is an element on $F$. Any uniform filter $F$ is contained in a minimal Gabriel filter $G$ (said to be generated by $F$), namely the intersection of all Gabriel filters containing $F$. Given a Gabriel filter $G$, the class of all $G$-torsion modules (see uniform filter) is a hereditary torsion class.

Literature

• Pierre Gabriel, Des catégories abéliennes
• Bo Stenstroem?, Rings of quotients, Springer 1975.
• Zoran Škoda, Noncommutative localization in noncommutative geometry, London Math. Society Lecture Note Series 330, ed. A. Ranicki; pp. 220–313, math.QA/0403276.
• Wendy Lowen, A generalization of the Gabriel–Popescu theorem, J. Pure Appl. Alg. 190:1–3 (2004) 197-211 doi

A version for quantales:

• Kimmo I. Rosenthal, A general approach to Gabriel filters on quantales, Comm. Algebra 20, n.11 (1992) 3393–3409 doi