$τ_d$-tilting theory for linear Nakayama algebras

TL;DR

The paper develops a comprehensive higher-dimensional extension of τ-tilting theory for truncated linear Nakayama algebras $\Lambda(n,l)$ that admit a $d$-cluster tilting subcategory. It provides explicit combinatorial classifications of summand-maximal $\tau_d$-rigid pairs $(M,P)$ with $|M|+|P|=n$, describes all $d$-torsion classes via graphs and admissible configurations, and links these to $(d+1)$-term silting complexes through thick-closure arguments. A central innovation is the notions of admissible configurations and well-configured $\mathcal{C}$-pairs, which enable a local-to-global analysis across the diagonals of the $d$-cluster tilting subcategory; this yields mutation results and a silting-theoretic perspective that extends classical theory (case $d=1$) to higher $d$. The work further analyzes special cases, including $l=2$ and $d=\mathrm{gl.dim.}(\Lambda)$, and provides algorithmic descriptions for constructing $d$-torsion classes and counting maximal $\tau_d$-rigid pairs, with implications for the lattice of $d$-torsion classes and mutation graphs.

Abstract

Support $τ$-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 $τ_d$-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 $Λ(n,l)=\mathbf{k} \mathbb{A}_{n}/\mathrm{rad}{\mathbf{k} \mathbb{A}_{n}}^l$ admitting a $d$-cluster tilting module. More specifically, we classify $τ_d$-rigid pairs $(M,P)$ of $Λ(n,l)$ 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 $\mathrm{K}^b(\mathrm{proj}(Λ(n,l)))$. We also describe all $d$-torsion classes of $Λ(n,l)$. 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 $Λ$.

Figures (1)

• Figure 1: The sm $\tau_2^{\space}$-rigid graph of $\Lambda(9,3)$.

Theorems & Definitions (150)

• Theorem A: Theorem \ref{['thrm:taud tilting is well-configured']}
• Theorem B: Theorem \ref{['thrm:classification of d-torsion classes']} and Theorem \ref{['thrm:classification of d-torsion classes in l=2']}
• Theorem C: Theorem \ref{['Theorem:strongly maximal and silting']}
• Definition 2.1
• Theorem 2.2: iyama2008auslander
• Definition 2.3
• Lemma 2.4: ADACHI2016227
• Lemma 2.5: VASO20192101
• Theorem 2.6: VASO20192101
• Remark 2.7