Localization on certain Grothendieck categories (Q1034263)

This paper is devoted to study TTF-classes in a Grothendieck category \(\mathcal A\) with a set of small projective generators. Recall that a non-empty class of objects \(\mathcal C\) of \(\mathcal A\) is said to be localizing if it is closed under subobjects, quotient objects, extensions and arbitrary direct sums. The localizing subcategory \(\mathcal C\) is called a TTF-class if, in addition, \(\mathcal C\) is closed under direct products. The authors prove that it suffices for a localizing subcategory of a semiartinian Grothendieck category to be a TTF-class, that every simple object has a projective cover. One of the main results of the paper (Theorem 3.2) states that the canonical functor \(T:\mathcal A\to \mathcal A/\mathcal C\) (where \(\mathcal C\) is a localizing subcategory of \(\mathcal A\)) has a left adjoint functor if and only if \(\mathcal C\) is a TTF-class. As application, they give sufficient conditions for a Grothendieck category \(\mathcal A\) with a set of small generating projective objects to be semiartinian, as well as they characterize when \(\mathcal A\) is semiperfect.