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