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Strongly tilting truncated path algebras

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eScholarship - University of California
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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 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 $${\mathcal{P}^{<\infty}({\rm Mod}\text{-}\tilde{\Lambda})}$$ .

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