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, Volume 2 (2019) no. 5, pp. 879-901.

Abstract

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 $A_{n}$ .

Received: 2018-05-18
Revised: 2019-01-14
Accepted: 2019-03-06
Published online: 2019-10-08

MR Zbl

DOI: 10.5802/alco.72

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

Author's affiliations:

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

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Barnard, Emily and Carroll, Andrew and Zhu, Shijie},
     title = {Minimal inclusions of torsion classes},
     journal = {Algebraic Combinatorics},
     pages = {879--901},
     publisher = {MathOA foundation},
     volume = {2},
     number = {5},
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     doi = {10.5802/alco.72},
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Barnard, Emily; Carroll, Andrew; Zhu, Shijie. Minimal inclusions of torsion classes. Algebraic Combinatorics, Volume 2 (2019) no. 5, pp. 879-901. doi : 10.5802/alco.72. https://alco.centre-mersenne.org/articles/10.5802/alco.72/

References
Cited by

[1] Adachi, Takahide; Iyama, Osamu; Reiten, Idun $τ$ -tilting theory, Compos. Math., Volume 150 (2014) no. 3, pp. 415-452 | DOI | MR | Zbl

[2] Asai, Sota Semibricks (2018) (to appear in Int. Math. Res. Not, https://arxiv.org/abs/1610.05860)

[3] Assem, Ibrahim; Skowroński, Andrzej Iterated tilted algebras of type ${\tilde{A}}_{n}$ , Math. Z., Volume 195 (1987) no. 2, pp. 269-290 | DOI | MR | Zbl

[4] Auslander, Maurice; Reiten, Idun; Smalø, Sverre O. Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, 36, Cambridge University Press, Cambridge, 1997, xiv+425 pages (Corrected reprint of the 1995 original) | MR

[5] Barnard, Emily The canonical join complex, Electronic J. Combinatorics, Volume 26 (2019) no. 1, Paper no. P1.24, 25 pages | MR | Zbl

[6] Brüstle, Thomas; Smith, David; Treffinger, Hipolito Wall and Chamber Structure for finite-dimensional Algebras (2018) (Preprint available: https://arxiv.org/abs/1805.01880) | Zbl

[7] Butler, Michael C. R.; Ringel, Claus Michael Auslander–Reiten sequences with few middle terms and applications to string algebras, Comm. Algebra, Volume 15 (1987) no. 1-2, pp. 145-179 | DOI | MR | Zbl

[8] Caspard, Nathalie; Le Conte de Poly-Barbut, Claude; Morvan, Michel Cayley lattices of finite Coxeter groups are bounded, Adv. in Appl. Math., Volume 33 (2004) no. 1, pp. 71-94 | DOI | MR | Zbl

[9] Crawley-Boevey, William Maps between representations of zero-relation algebras, J. Algebra, Volume 126 (1989) no. 2, pp. 259-263 | DOI | MR | Zbl

[10] Demonet, Laurent; Iyama, Osamu; Jasso, Gustavo $τ$ -tilting finite algebras, bricks, and $g$ -vectors, Int. Math. Res. Not. IMRN (2019) no. 3, pp. 852-892 | DOI | MR | Zbl

[11] Demonet, Laurent; Iyama, Osamu; Reading, Nathan; Reiten, Idun; Thomas, Hugh Lattice theory of torsion classes (2018) (Preprint available: https://arxiv.org/abs/1711.01785)

[12] Garver, Alexander; McConville, Thomas Lattice Properties of Oriented Exchange Graphs and Torsion Classes, Algebr. Represent. Theory, Volume 22 (2019) no. 1, pp. 43-78 | DOI | MR | Zbl

[13] Iyama, Osamu; Reading, Nathan; Reiten, Idun; Thomas, Hugh Lattice structure of Weyl groups via representation theory of preprojective algebras, Compos. Math., Volume 154 (2018) no. 6, pp. 1269-1305 | DOI | MR | Zbl

[14] Iyama, Osamu; Reiten, Idun; Thomas, Hugh; Todorov, Gordana Lattice structure of torsion classes for path algebras, Bull. Lond. Math. Soc., Volume 47 (2015) no. 4, pp. 639-650 | DOI | MR | Zbl

[15] Mizuno, Yuya Classifying $τ$ -tilting modules over preprojective algebras of Dynkin type, Math. Z., Volume 277 (2014) no. 3-4, pp. 665-690 | DOI | MR | Zbl

[16] Reading, Nathan Noncrossing arc diagrams and canonical join representations, SIAM J. Discrete Math., Volume 29 (2015) no. 2, pp. 736-750 | DOI | MR | Zbl

[17] Smalø, Sverre O. Torsion theories and tilting modules, Bull. London Math. Soc., Volume 16 (1984) no. 5, pp. 518-522 | DOI | MR | Zbl

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