$\tau_d$-tilting theory for linear Nakayama algebras

TL;DR

This paper extends $ au$-tilting theory to higher dimensions for linear Nakayama algebras, classifying $ au_d$-rigid pairs and $d$-torsion classes, and exploring their connections and mutations.

Contribution

It provides a combinatorial classification of $ au_d$-rigid pairs and describes $d$-torsion classes for truncated linear Nakayama algebras, extending classical $ au$-tilting theory.

Findings

Classified $ au_d$-rigid pairs explicitly.

Characterized $ au_d$-rigid pairs via maximality and silting complexes.

Described all $d$-torsion classes for $ abla(n,l)$.

Abstract

Support $\tau$-tilting pairs, functorially finite torsion classes and $2$-term silting complexes are three much studied concepts in the representation theory of finite-dimensional algebras, which moreover turn out to be connected via work of Adachi, Iyama and Reiten. We investigate their higher-dimensional analogues via $\tau_d$-rigid pairs, $d$-torsion classes and $(d+1)$-term silting complexes as well as the connections between these three concepts. Our work is done in the setting of truncated linear Nakayama algebras $\Lambda(n,l)=\mathbf{k} \mathbb{A}_{n}/\mathrm{rad}{\mathbf{k} \mathbb{A}_{n}}^l$ admitting a $d$-cluster tilting module. More specifically, we classify $\tau_d$-rigid pairs $(M,P)$ of $\Lambda(n,l)$ with $|M|+|P|=n$ via an explicit combinatorial description and show that they can be characterized by a certain maximality condition as well as by giving rise to a $(d+1)$-term silting complex in $\mathrm{K}^b(\mathrm{proj}(\Lambda(n,l)))$. We also describe all $d$-torsion classes of $\Lambda(n,l)$. Finally, we compare our results to the classical case $d=1$ and investigate mutation with a special emphasis on the case where $d$ equals the global dimension of $\Lambda$.