# ALGEBRAIC COMBINATORICS

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

Let $\Lambda$ be a finite-dimensional associative algebra. The torsion classes of $\mathrm{mod}\Lambda$ form a lattice under containment, denoted by $\mathrm{tors}\Lambda$. In this paper, we characterize the cover relations in $\mathrm{tors}\Lambda$ 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 $\mathrm{tors}\Lambda$ in terms of representation theory. Finally, we show that, in general, the algebra $\Lambda$ is not characterized by its lattice $\mathrm{tors}\Lambda$. 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 ${A}_{n}$.

Revised: 2019-01-14
Accepted: 2019-03-06
Published online: 2019-10-08
DOI: https://doi.org/10.5802/alco.72
Classification: 05E10,  06B15
Keywords: lattice theory, torsion classes, canonical join representations
@article{ALCO_2019__2_5_879_0,
author = {Barnard, Emily and Carroll, Andrew and Zhu, Shijie},
title = {Minimal inclusions of torsion classes},
journal = {Algebraic Combinatorics},
publisher = {MathOA foundation},
volume = {2},
number = {5},
year = {2019},
pages = {879-901},
doi = {10.5802/alco.72},
mrnumber = {4023570},
zbl = {1428.05314},
language = {en},
url = {alco.centre-mersenne.org/item/ALCO_2019__2_5_879_0/}
}
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/item/ALCO_2019__2_5_879_0/

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