Strongly tilting truncated path algebras

Abstract

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