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GABRIEL FILTERS IN GROTHENDIECK CATEGORIES

A. Jeremías López, M. P. López López and E. Villanueva Nóvoa
Publicacions Matemàtiques
Vol. 36, No. 2A (1992), pp. 673-683
Stable URL: http://www.jstor.org/stable/43736396
Page Count: 11
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GABRIEL FILTERS IN GROTHENDIECK CATEGORIES
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Abstract

In [1] it is proved that one must take care trying to copy results from the case of modules to an arbitrary Grothendieck category in order to describe a hereditary torsion theory in terms of filters of a generator. By the other side, we usually have for a Grothendieck category an infinite family of generators {Gij i ∊ I} and, although each Gi has good properties the generator $G = _{i \in I}^ \oplus {G_i}$is not easy to handle (for instance in categories like graded modules, presheaves or sheaves of modules). In this paper the authors obtain a bijective correspondence between hereditary torsion theories in a Grothendieck category C and a appropriately defined family of Gabriel filters of subobjets of the generators of C. This has been possible by using the natural conditions of local projectiveness and local smallness for families of generators in a Grothendieck category, that the embedding theorem of Gabriel-Popescu provided us.

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