Infinite $\tau$-tilting theory

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 $\tau$-tilting theory, where torsion classes correspond to support $\tau$-tilting subsets of the Ziegler spectrum and torsion-free classes correspond to support $\tau^-$-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 $\tau^-$-rigid modules, in particular for tame algebras, for which we show that these classes of modules coincide. We also recover a result of Bautista, P\'erez and Salmer\'on 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.