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

摘要

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(lR) and lR is the Bongartz complement of P modulo its P-trace. Moreover, there is a ring epimorphism, phi : R -> S. We characterize the case when phi 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 mu' and nu' between the interval [GenP, P-L I I 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 mu' and nu' 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 mu' and nu, contains all injectives. (c) 2007 Elsevier B.V. All rights reserved.