Strongly tilting truncated path algebras

For any truncated path algebra Λ, we give a structural description of the modules in the categories {\mathcal{P}^{<\infty}(\Lambda\text{-}{\rm mod})} and {\mathcal{P}^{<\infty}(\Lambda\text{-}{\rm mod})} , consisting of the finitely generated (resp. arbitrary) Λ-modules of finite projective dimension. We deduce that these categories are contravariantly finite in Λ−mod and Λ-Mod, respectively, and determine the corresponding minimal {\mathcal{P}^{<\infty}} -approximation of an arbitrary Λ-module from a projective presentation. In particular, we explicitly construct—based on the Gabriel quiver Q and the Loewy length of Λ—the basic strong tilting module Λ T (in the sense of Auslander and Reiten) which is coupled with {\mathcal{P}^{<\infty}(\Lambda\text{-}{\rm mod})} in the contravariantly finite case. A main topic is the study of the homological properties of the corresponding tilted algebra $${\tilde{\Lambda} = {\rm End}_\Lambda(T)^{\rm op}}$$ , such as its finitistic dimensions and the structure of its modules of finite projective dimension. In particular, we characterize, in terms of a straightforward condition on Q, the situation where the tilting module $${T_{\tilde{\Lambda}}}$$ is strong over $${\tilde{\Lambda}}$$ as well. In this Λ- $${\tilde{\Lambda}}$$ -symmetric situation, we obtain sharp results on the submodule lattices of the objects in {\mathcal{P}^{<\infty}({\rm Mod}\text{-}\tilde{\Lambda})} , among them a certain heredity property; it entails that any module in {\mathcal{P}^{<\infty}({\rm Mod}\text{-}\tilde{\Lambda})} is an extension of a projective module by a module all of whose simple composition factors belong to {\mathcal{P}^{<\infty}({\rm Mod}\text{-}\tilde{\Lambda})}