Higher torsion classes, $τ_d$-tilting theory and silting complexes
Jenny August, Johanne Haugland, Karin M. Jacobsen, Sondre Kvamme, Yann Palu, Hipolito Treffinger
TL;DR
This work extends classical $\tau$-tilting theory to a higher-dimensional setting via the $d$-Auslander–Reiten translation $\tau_d$, establishing connections between functorially finite $d$-torsion classes, maximal $\tau_d$-rigid pairs, and $(d+1)$-term silting complexes. It provides a robust framework that includes new results on $d$-cluster tilting subcategories in exact and triangulated contexts, a bijective approach to constructing Ext^d-projectives and $d$-tilting modules, and a silting-theoretic interpretation of higher torsion data. The paper also demonstrates concrete combinatorial descriptions for higher Auslander and higher Nakayama algebras and offers computational tools to generate explicit examples, while highlighting that certain maps are injective but not surjective in general for $d>1$. Overall, the results forge deep linkages among higher homological algebra, higher tilting theory, and silting theory, with implications for derived equivalences and higher cluster categories. The framework broadens the scope of mutation phenomena and provides explicit constructions of higher tilting/silting objects from $d$-torsion data, enriching both theory and computable practice.
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.