Tuesday, August, 14:00, 7.527.

Kevin Schlegel
Abstract:
We classify torsion pairs in an essentially small abelian category through cosilting subsets of the Ziegler spectrum of the ind-completion of the abelian category. For Artin algebras, this classification is reformulated as an infinite analog of τ-tilting theory, where torsion classes correspond to support τ-tilting subsets of the Ziegler spectrum and torsion-free classes correspond to support τ ^--tilting subsets. We further express the classification through ideals of the module category, thereby obtaining a formulation that involves finite length modules only. The developed theory is applied to study generic bricks and generic τ ^--rigid modules, in particular for tame algebras, for which we show that these classes of modules coincide. We also recover a result of Bautista, Pérez and Salmerón stating that a tame algebra admits infinitely many bricks of a fixed dimension if and only if there exists a generic brick. Finally, we prove that every algebra whose Krull-Gabriel dimension is defined satisfies the brick version of the second Brauer-Thrall conjecture.