Higher torsion classes, $τ_d$-tilting theory and silting complexes
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , , , , |
|---|---|
| Format: | Preprint |
| Publié: |
2026
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866910022906150912 |
|---|---|
| author | August, Jenny Haugland, Johanne Jacobsen, Karin M. Kvamme, Sondre Palu, Yann Treffinger, Hipolito |
| author_facet | August, Jenny Haugland, Johanne Jacobsen, Karin M. Kvamme, Sondre Palu, Yann Treffinger, Hipolito |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_03659 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Higher torsion classes, $τ_d$-tilting theory and silting complexes August, Jenny Haugland, Johanne Jacobsen, Karin M. Kvamme, Sondre Palu, Yann Treffinger, Hipolito Representation Theory Rings and Algebras 16S90, 18E40, 16G70, 16G20, 18G25, 18G99 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. |
| title | Higher torsion classes, $τ_d$-tilting theory and silting complexes |
| topic | Representation Theory Rings and Algebras 16S90, 18E40, 16G70, 16G20, 18G25, 18G99 |
| url | https://arxiv.org/abs/2602.03659 |