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Uniform filters. (English) Zbl 0936.16027

The notion of uniform (or “topologizing”) filter has been introduced by P. Gabriel [Bull. Soc. Math. Fr. 90, 323-448 (1962; Zbl 0201.35602)], who proved that idempotent uniform filters (nowadays referred to as “Gabriel filters”) over a ring R correspond bijectively to localizations of the category R-mod. At the later stage, O. Goldman [J. Algebra 13, 10-47 (1969; Zbl 0201.04002)] has pointed out that Gabriel filters are also in bijective correspondence with idempotent kernel functors.

In the past, uniform filters have mainly been considered within the framework of linear topologies. Recently, however, new applications of uniform filters arose in the context of noncommutative algebraic geometry. These applications require a deeper study of the functorial properties of uniform filters with respect to change of base ring and thus urged us to reconsider the notion of uniform filter.

In the first section of the paper under review, the authors recollect some general results on the lattice of uniform filters and examples of uniform filters associated to prime and arbitrary twosided ideals, naturally arisen in the framework of noncommutative algebraic geometry. In the second section, it is shown that the lattice of uniform filters possesses a quantale structure. The aim of this section is to associate, to any left R-module, a sheaf over this quantale with a suitably nice functorial behaviour and in the last section it is described how ring homomorphisms between rings R and S allow to induce well related uniform filters from R and S.

MSC:
16S90Torsion theories; radicals on module categories
18F20Categorical methods in sheaf theory
16S60Rings of functions, subdirect products, sheaves of rings
16S38Rings arising from non-commutative algebraic geometry