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- W2950724570 abstract "For any truncated path algebra $Lambda$, we give a structural description of the modules in the categories ${cal P}^{<infty}(Lambdatext{-mod})$ and ${cal P}^{<infty}(Lambdatext{-Mod})$, consisting of the finitely generated (resp. arbitrary) $Lambda$-modules of finite projective dimension. We deduce that these categories are contravariantly finite in $Lambdatext{-mod}$ and $Lambdatext{-Mod}$, 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 ${cal P}^{<infty}(Lambdatext{-mod})$ 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 ${cal P}^{<infty}(text{mod-}tilde{Lambda})$." @default.
- W2950724570 created "2019-06-27" @default.
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- W2950724570 date "2014-07-10" @default.
- W2950724570 modified "2023-09-27" @default.
- W2950724570 title "Strongly tilting truncated path algebras" @default.
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- W2950724570 doi "https://doi.org/10.48550/arxiv.1407.2690" @default.
- W2950724570 hasPublicationYear "2014" @default.
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