Minimal inclusions of torsion classes
Algebraic Combinatorics, Volume 2 (2019) no. 5, pp. 879-901.

Let Λ be a finite-dimensional associative algebra. The torsion classes of modΛ form a lattice under containment, denoted by torsΛ. In this paper, we characterize the cover relations in torsΛ 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 torsΛ in terms of representation theory. Finally, we show that, in general, the algebra Λ is not characterized by its lattice torsΛ. 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 .

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DOI: 10.5802/alco.72
Classification: 05E10,  06B15
Keywords: lattice theory, torsion classes, canonical join representations
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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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/

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