Correspondences from tilting theory in higher homological algebra
Adachi, Iyama and Reiten developed \(\tau\)-tilting theory to mirror the properties of mutation seen in cluster algebras. The theory gives a generalisation of classical tilting modules using the Auslander-Reiten translation \(\tau\), and one studies distinguished pairs of objects in the module category of a finite-dimensional algebra known as \(\tau\)-rigid pairs. An important result from the theory is the correspondence between functorially finite torsion classes, maximal \(\tau\)-rigid pairs and \(2\)-term silting complexes, amongst others.
Meanwhile, higher homological algebra has since its introduction by Iyama become a very active field of research, and many authors have generalised notions to the higher homological setting, including both torsion classes and \(\tau\)-rigid pairs. This talk is a report on work in progress investigating the relationship between higher torsion classes, silting objects and maximal \(\tau_d\)-rigid pairs. We describe explicit correspondences, and also show computational results.
The talk is based on joint work with August, Haugland, Kvamme, Palu and Treffinger