Minimal inclusions of torsion classes
Algebraic Combinatorics, Volume 2 (2019) no. 5, p. 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 .

Received : 2018-05-18
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},
     language = {en},
     url = {https://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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