Minimal inclusions of torsion classes

By Emily Barnard, Andrew T. Carroll, Shijie Zhu. Algebraic Combinatorics, 2017

77 citations recorded by the source corpus.

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

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b8a2700be11246c1f6d01099429fa86c90aeaf24

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

A survey on maximal green sequences (2019, background): “nsional algebra and I ŸA a 2-sided ideal. Then the map TÞÑTXmodpA{Iq induces a contraction HasseptorsAq Ñ HasseptorsA{Iq. Proof. Let us ﬁrst recall from Theorem 3.3 of [33], cf. also Theorem 1.0.2 of [6], that an inclusion of torsion classes SĎ T is minimal if and only if there is a unique (up to is”
Wide Subcategories and Lattices of Torsion Classes (2019, methodology): “s the isomorphism class of the brick Sq from Proposition 3.2. In this case, we write T −→ US q . The brick labeling may be compared to minimal extending modules for U as deﬁned in Barnard–Carroll–Zhu [BCZ]. The minimal extending modules for U are precisely the labels of arrows starting at U, see [BC”
Wall and chamber structure for finite-dimensional algebras (2018, background): “This labeling by bricks appeared independently in [2, 16, 7].”