Vol. 298, No. 2, 2019

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Tilting modules over Auslander–Gorenstein algebras

Osamu Iyama and Xiaojin Zhang

Vol. 298 (2019), No. 2, 399–416
Abstract

For a finite-dimensional algebra Λ and a nonnegative integer n, we characterize when the set  tiltnΛ of additive equivalence classes of tilting modules with projective dimension at most n has a minimal (or equivalently, minimum) element. This generalizes results of Happel and Unger. Moreover, for an n-Gorenstein algebra Λ with n 1, we construct a minimal element in  tiltnΛ. As a result, we give equivalent conditions for a k-Gorenstein algebra to be Iwanaga–Gorenstein. Moreover, for a 1-Gorenstein algebra Λ and its factor algebra Γ = Λ(e), we show that there is a bijection between  tilt1Λ and the set  sτ -tilt Γ of additive equivalence classes of basic support τ-tilting Γ-modules, where e is an idempotent such that eΛ is the additive generator of the category of projective-injective Λ-modules.

Keywords
$n$-Gorenstein algebra, tilting module, support $\tau$-tilting module
Mathematical Subject Classification 2010
Primary: 16G10
Secondary: 16E10
Milestones
Received: 19 March 2018
Revised: 8 July 2018
Accepted: 14 July 2018
Published: 8 March 2019
Authors
Osamu Iyama
Graduate School of Mathematics
Nagoya University
Chikusa-ku
Nagoya
Japan
Xiaojin Zhang
School of Mathematics and Statistics
Nanjing University of Information Science and Technology
Nanjing
China