On the exactness of products in the localization of (Ab.\(4^{\ast}\)) Grothendieck categories (Q335589)

Let \(\mathcal G\) be a Grothendieck category satisfying the axiom AB4* (this means that arbitrary direct products do exist in \(\mathcal G\) and any direct product of epimorphisms in \(\mathcal G\) is again an epimorphism). The aim of this paper is to investigate the exactness of direct products in the quotient categories \(\mathcal{G}/\mathcal{T}\) of \(\mathcal G,\) where \(\mathcal T\) is a localizing subcategory (i.e., a hereditary torsion class of \(\mathcal G\)). Recall that a Grothendieck category \(\mathcal G\) is said to be \textit{stable} if \(\mathcal T\) is closed under taking injective envelopes for any localizing subcategory \(\mathcal T\) of \(\mathcal G.\) The author says that a Grothendieck category \(\mathcal G\) is \textit{effective} if it is a locally Noetherian, stable category such that all its prime localizing subcategories are exact.NEWLINENEWLINEAmong others, the author proves that if \(\mathcal G\) is an effective Grothendieck category satisfying the axiom AB4*, then the \((k+1)\)-th derived functor of the product vanishes in case the Gabriel dimension of the quotient category \(\mathcal{G}/\mathcal{T}\) is \(\,k<\infty\); as a corollary, the author shows that, in this case, the derived category \(\mathbb{D}(\mathcal{G}/\mathcal{T})\) is left-complete.