Dualities from iterated tilting
type: Item
instance of: scholarly article
title: Dualities from iterated tilting
zbMATH Open document ID: 1483.16016
author: B. Huisgen-Zimmermann
published in: Israel Journal of Mathematics
publication date: 2021-08-24T00:00:00Z
review text: 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 Nu…
reviewed by: Alexis Langlois-Rémillard
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 16G20
zbMATH DE Number: 7384530
zbMATH Keywords: truncated path algebras
zbMATH Keywords: strong tilting modules
zbMATH Keywords: modules of finite projective dimension
zbMATH Keywords: categories of modules
zbMATH Keywords: dualities
MaRDI profile type: Publication
full work available at URL: s11856-021-2161-x
OpenAlex ID: W3166672654
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DOI: 10.1007/S11856-021-2161-X
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description: scientific article; zbMATH DE number 7384530