Perpendicular categories of infinite dimensional partial tilting modules and transfers of tilting torsion classes

Let RR be a ring and PP be an (infinite dimensional) partial tilting module. We show that the perpendicular category of PP is equivalent to the full module category Mod-S where S=End(ℓR) and ℓRℓR is the Bongartz complement of PP modulo its PP-trace. Moreover, there is a ring epimorphism φ:R→Sφ:R→S. We characterize the case when φφ is a perfect localization. By [Riccardo Colpi, Alberto Tonolo, Jan Trlifaj, Partial cotilting modules and the lattices induced by them, Comm. Algebra 25 (10) (1997) 3225–3237], there exist mutually inverse isomorphisms μ′μ′ and ν′ν′ between the interval [GenP,P⊥1] in the lattice of torsion classes in Mod-R, and the lattice of all torsion classes in Mod-S. We provide necessary and sufficient conditions for μ′μ′ and ν′ν′ to preserve tilting torsion classes. As a consequence, we show that these conditions are always satisfied when RR is a Dedekind domain, and if PP is finitely presented and RR is an artin algebra, then the conditions reduce to the trivial ones, namely that each value of μ′μ′ and ν′ν′ contains all injectives.