Minimal inclusions of torsion classes

Barnard, Emily; Carroll, Andrew; Zhu, Shijie

doi:10.5802/alco.72

Barnard, Emily ¹ ; Carroll, Andrew ² ; Zhu, Shijie ³

¹ Department of Mathematical Sciences DePaul University 2320 N. Kenmore Ave. Suite 502 Chicago IL 60614, USA
² 3778 Keating St. San Diego CA 92110, USA
³ Mathematics Department University of Iowa 14 MacLean Hall Iowa City IA 52242, USA

Algebraic Combinatorics, Tome 2 (2019) no. 5, pp. 879-901

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Résumé

Let $Λ$ be a finite-dimensional associative algebra. The torsion classes of $\mod Λ$ form a lattice under containment, denoted by $tors Λ$ . In this paper, we characterize the cover relations in $tors Λ$ by certain indecomposable modules. We consider three applications: First, we show that the completely join-irreducible torsion classes (torsion classes which cover precisely one element) are in bijection with bricks. Second, we characterize faces of the canonical join complex of $tors Λ$ in terms of representation theory. Finally, we show that, in general, the algebra $Λ$ is not characterized by its lattice $tors Λ$ . In particular, we study the torsion theory of a quotient of the preprojective algebra of type $A_{n}$ . We show that its torsion class lattice is isomorphic to the weak order on $A_{n}$ .

Reçu le : 2018-05-18
Révisé le : 2019-01-14
Accepté le : 2019-03-06
Publié le : 2019-10-08

MR Zbl

DOI : 10.5802/alco.72

Classification : 05E10, 06B15
Keywords: lattice theory, torsion classes, canonical join representations

Affiliations des auteurs :

Barnard, Emily ¹ ; Carroll, Andrew ² ; Zhu, Shijie ³

¹ Department of Mathematical Sciences DePaul University 2320 N. Kenmore Ave. Suite 502 Chicago IL 60614, USA
² 3778 Keating St. San Diego CA 92110, USA
³ Mathematics Department University of Iowa 14 MacLean Hall Iowa City IA 52242, USA

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Minimal inclusions of torsion classes},
     journal = {Algebraic Combinatorics},
     pages = {879--901},
     publisher = {MathOA foundation},
     volume = {2},
     number = {5},
     year = {2019},
     doi = {10.5802/alco.72},
     mrnumber = {4023570},
     zbl = {1428.05314},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.5802/alco.72/}
}

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Barnard, Emily; Carroll, Andrew; Zhu, Shijie. Minimal inclusions of torsion classes. Algebraic Combinatorics, Tome 2 (2019) no. 5, pp. 879-901. doi: 10.5802/alco.72

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