Strongly tilting truncated path algebras

Abstract

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 ΛT (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 \({\mathcal{P}^{<\infty}({\rm Mod}\text{-}\tilde{\Lambda})}\) .