Minimal inclusions of torsion classes

Abstract

Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> </mml:math> be a finite-dimensional associative algebra. The torsion classes of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="normal">mod</mml:mi> <mml:mi>Λ</mml:mi> </mml:mrow> </mml:math> form a lattice under containment, denoted by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="normal">tors</mml:mi> <mml:mi>Λ</mml:mi> </mml:mrow> </mml:math> . In this paper, we characterize the cover relations in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="normal">tors</mml:mi> <mml:mi>Λ</mml:mi> </mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="normal">tors</mml:mi> <mml:mi>Λ</mml:mi> </mml:mrow> </mml:math> in terms of representation theory. Finally, we show that, in general, the algebra <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> </mml:math> is not characterized by its lattice <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="normal">tors</mml:mi> <mml:mi>Λ</mml:mi> </mml:mrow> </mml:math> . In particular, we study the torsion theory of a quotient of the preprojective algebra of type <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> . We show that its torsion class lattice is isomorphic to the weak order on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> .