Pure mathematics

Biology

Discrete mathematics

Surgery

Computer science

Acoustics

Lattice (music)

Associative property

Torsion (gastropod)

Physics

Anatomy

Mathematics

Bijection

Medicine

Indecomposable module

Quotient

Algebra over a field

Algebraic Combinatorics

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 .

Minimal inclusions of torsion classes