Dualities from iterated tilting
This paper follows two goals. First it studies dualities for tilting modules in the full subcategory \(\mathcal{P}^{<\infty}(\Lambda\text{-mod})\) of modules of finite projective dimension over a finite-dimensional algebra \(\Lambda\). For \(T\), a tilting module of \(\Lambda\) and \(\widetilde{\Lambda}\), its corresponding tilted algebra, there is a duality coming from the work of \textit{Y. Miyashita} [Math. Z. 193, 113--146 (1986; Zbl 0578.16015)] between the two contravariant functors \[ \operatorname{Hom}_{\Lambda}(-,T) : \mathcal{P}^{<\infty}(\Lambda\text{-mod}) \longleftrightarrow \mathcal{P}^{<\infty}(\text{mod-}\widetilde{\Lambda}) : \operatorname{Hom}_{\widetilde{\Lambda}}(-,T) \] when \( _\Lambda T_{\widetilde{\Lambda}}\) is a tilting bimodule strong on both side, so if it is Ext-injective relative to objects of \(\mathcal{P}^{<\infty}(\Lambda\text{-mod})\) and \(\mathcal{P}^{<\infty}(\text{mod-}\widetilde{\Lambda})\). Theorem 1 of the article proves a converse statement in a more general context whose corollary is that, if there is such a pair of dualities between two finite-dimensional algebras, then they can be induced by a tilting bimodule strong on both sides. The article then specialises its investigation to truncated path algebras. On this it follows a second goal: to exhibit the special behavior of these algebra in relation to tilting and compares them with hereditary algebras at the homological level. Notably it provides statements on projective dimension of modules in the bigger category \(\text{Mod-}\widetilde{\Lambda}\) (Theorem 12), on the passage of contravariant finiteness from \(\mathcal{P}^{<\infty}(\Lambda\text{-mod})\) to \(\mathcal{P}^{<\infty}(\text{mod-}\widetilde{\Lambda})\) (Theorem 17) and on the 2-periodicity of iterated strong tilting (Theorem 19). This study builds upon previous works [\textit{A. Dugas} and the author, Manuscr. Math. 134, No. 1--2, 225--257 (2011; Zbl 1231.16013); \textit{K. Goodearl} and the author, Algebra Number Theory 12, No. 2, 379--410 (2018; Zbl 1420.16008)] and its applications are illustrated in later work [the author and \textit{M. Saorín}, Contemp. Math. 761, 61--101 (2021; Zbl 1470.16025)]. A nice feature of the article is example (3B) used through the text to ground the results.
scientific article; zbMATH DE number 7384530
scientific article