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 Grothendick category an infinite family of generators {Gi; i ∈ I} and, although each Gi has good properties the generator G = ⊕i ∈ I Gi is not easy to handle (for instance in categories like graded modules). In this paper the authors obtain a bijective correspondence between hereditary torsion theories in a Grothendieck category C and an appropriately defined family of Gabriel filters of subobjects 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 thorem of Gabril-Popescu provided us.
@article{urn:eudml:doc:41744,
title = {Gabriel filters in Grothendieck categories.},
journal = {Publicacions Matem\`atiques},
volume = {36},
year = {1992},
pages = {673-683},
mrnumber = {MR1209832},
zbl = {0778.18010},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41744}
}
Jeremías López, Ana; López López, María Purificación; Villanueva Nóvoa, Emilio. Gabriel filters in Grothendieck categories.. Publicacions Matemàtiques, Tome 36 (1992) pp. 673-683. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41744/