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

Abstract

Let R be a ring and P be an (infinite dimensional) partial tilting module. We show that the perpendicular category of P is equivalent to the full module category Mod - S where S = End ( ℓ R ) and ℓ R is the Bongartz complement of P modulo its P -trace. Moreover, there is a ring epimorphism φ : R → S . We characterize the case when φ 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 ′ and ν ′ between the interval [ Gen P , 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 ′ and ν ′ to preserve tilting torsion classes. As a consequence, we show that these conditions are always satisfied when R is a Dedekind domain, and if P is finitely presented and R is an artin algebra, then the conditions reduce to the trivial ones, namely that each value of μ ′ and ν ′ contains all injectives.