On relative Auslander algebras. (English) Zbl 1436.16019
The paper deals with relative Auslander algebras. For a given Artinian algebra \(A\), let \(\mathscr{X}\) be a contravariantly finite full subcategory of \({}^\perp A\subset\mathbf{mod}(A)\) containing the projectives. Assume that \(\mathscr{X}\) contains an additive generator \(X\). Its endomorphism ring \(\Gamma\) is the relative Auslander algebra. The authors exhibit a \(\Gamma\)-module which is shown to be a tilting module if \(\mathscr{X}\) is closed under syzygies, and a cotilting module if \(\mathscr{X}\) is also closed under submodules.
This theorem generalizes several known results, including Cohen-Macaulay Auslander algebras and \(n\)-Auslander algebras (see [O. Iyama and Ø. Solberg, Adv. Math. 326, 200–240 (2018; Zbl 1432.16012)]). The “gentle” hypothesis in X. Chen and M. Lu’s paper on gentle algebras [Algebr. Represent. Theory 19, No. 6, 1321–1345 (2016; Zbl 1379.13004)] is shown to be redundant. Furthermore, it is shown that an algebra \(A\) is representation-finite whenever it has a relative Auslander algebra of finite type.
This theorem generalizes several known results, including Cohen-Macaulay Auslander algebras and \(n\)-Auslander algebras (see [O. Iyama and Ø. Solberg, Adv. Math. 326, 200–240 (2018; Zbl 1432.16012)]). The “gentle” hypothesis in X. Chen and M. Lu’s paper on gentle algebras [Algebr. Represent. Theory 19, No. 6, 1321–1345 (2016; Zbl 1379.13004)] is shown to be redundant. Furthermore, it is shown that an algebra \(A\) is representation-finite whenever it has a relative Auslander algebra of finite type.
Reviewer: Wolfgang Rump (Stuttgart)
MSC:
| 16G60 | Representation type (finite, tame, wild, etc.) of associative algebras |
| 16G50 | Cohen-Macaulay modules in associative algebras |
| 18A25 | Functor categories, comma categories |
| 18G25 | Relative homological algebra, projective classes (category-theoretic aspects) |
| 16S50 | Endomorphism rings; matrix rings |
| 16S90 | Torsion theories; radicals on module categories (associative algebraic aspects) |
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