Strongly tilting truncated path algebras

TL;DR

This paper provides a detailed structural analysis of modules with finite projective dimension over truncated path algebras, constructs strong tilting modules explicitly, and examines the homological properties of the associated tilted algebras.

Contribution

It explicitly constructs strong tilting modules for truncated path algebras and analyzes their homological properties and submodule structures in the tilted algebra context.

Findings

Categories of modules with finite projective dimension are contravariantly finite.

Explicit construction of the basic strong tilting module based on Gabriel quiver and Loewy length.

Characterization of when the tilting module is strong over the tilted algebra.

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 ${\cal P}^{<\infty}(\text{mod-}\tilde{\Lambda})$.