Given an hereditary torsion theory $(\mathcal{T}, \mathcal{F})$ on an abelian category $\mathcal{A}$, how we can relate this to a localization (i.e Ore localization). This is mentioned with not so much detail or with a lot of encyclopedic references (hard to follow), in a lot of books and papers concerning torsion theory. Still dont see how given an hereditary torsion theory $(\mathcal{T}, \mathcal{F})$ we can produce a subclass of morphism $W \subset \mathcal{A}$ such that there is a pair $(T, C_{W})$ where $T:C \to C_{W}$ is a universal functor and $C_{W}$ is at least an additive category. And where $T(w)$ is and isomorphism in $C_{W}$ for each morphism $w \in W$. Also, viceversa, given a localization like mentioned before...How we can produce a torsion theory?
Also when $T$ is a triangulated torsion and $(\mathcal{T}, \mathcal{F})$ is an hereditary torsion in a triangulated category how can we produce a localization of $T$ over subclass of morphisms $W \subset Mor(T)$ like the one mentioned before and $T_{W}$ preserves the triangulated structure of $T$. The same doubt for the converse process.
