Higher torsion classes, $\tau_d$-tilting theory and silting complexes

TL;DR

This paper extends $ au$-tilting theory to higher dimensions using $ au_d$, establishing new bijections, complexes, and examples in higher Auslander-Reiten theory, with applications to $d$-APR tilting modules and higher Nakayama algebras.

Contribution

It introduces a higher-dimensional analogue of $ au$-tilting theory, associating functorially finite $d$-torsion classes with maximal $ au_d$-rigid pairs and $(d+1)$-term silting complexes, and explores their properties.

Findings

Established bijections between $d$-torsion classes and silting complexes.

Produced explicit combinatorial descriptions for higher Auslander and Nakayama algebras.

Provided new examples of $d$-cluster tilting subcategories.

Abstract

Initiated in work by Adachi, Iyama and Reiten, the area known as $\tau$-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 $\tau_d$. In particular, we associate to any functorially finite $d$-torsion class a maximal $\tau_d$-rigid pair and a $(d+1)$-term silting complex. In the case $d=1$, the notions of maximal $\tau_d$-rigid and support $\tau$-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 $\tau_d$-rigid pairs and $(d+1)$-term silting complexes for higher Auslander and higher Nakayama algebras.