On the exactness of products in the localization of (Ab.4⁎) Grothendieck categories

Let CC be an (Ab.4⁎) Grothendieck category, that is, products are exact in CC. Given a hereditary torsion class T⊆CT⊆C, we study the exactness of products in the Gabriel localization C/TC/T of CC. We show that, under suitable assumptions on CC, the k+1k+1-th derived functor of the product vanishes, provided the Gabriel dimension of C/TC/T is smaller than k  . As a consequence, we deduce that, under suitable hypotheses, the derived category D(C/T)D(C/T) is left-complete.