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.
Perpendicular categories of infinite dimensional partial tilting modules and transfers of tilting torsion classes
COLPI, RICCARDO;TONOLO, ALBERTO;
2007
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(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.Pubblicazioni consigliate
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