Higher torsion classes, $τ_d$-tilting theory and silting complexes

Abstract

Initiated in work by Adachi, Iyama and Reiten, the area known as $τ$-tilting theory plays a fundamental role in contemporary representation theory. In this paper we explore a higher-dimensional analogue of this theory, formulated with respect to the higher Auslander-Reiten translation $τ_d$. In particular, we associate to any functorially finite $d$-torsion class a maximal $τ_d$-rigid pair and a $(d+1)$-term silting complex. In the case $d=1$, the notions of maximal $τ_d$-rigid and support $τ$-tilting pairs coincide, and our theory recovers the classical bijections. However, the proof strategies for $d>1$ differ significantly. As an intermediate step, we prove that a $d$-cluster tilting subcategory of a module category induces a $d$-cluster tilting subcategory of the category of $(d+1)$-term complexes, producing novel examples of $d$-exact categories. We introduce the notion of a $d$-torsion class in the exact setup, and use this to obtain the aforementioned $(d+1)$-term silting complex. We moreover apply our theory to study $d$-APR tilting modules and slices. To illustrate our results, we provide explicit combinatorial descriptions of maximal $τ_d$-rigid pairs and $(d+1)$-term silting complexes for higher Auslander and higher Nakayama algebras.