Relative delooping levels on intervals of torsion classes
Abstract
Let U be a support τ-tilting module over a finite-dimensional algebra A, and let 𝒱⊆FacU be any torsion class. The exact heart ℋ=FacU∩𝒱⊥ has an Ext-progenerator P. We prove that HomA(P,−) realizes ℋ as a submodule-closed exact subcategory of modB, where B=EndA(P), and preserves minimal syzygies. The global relative k-delooping levels are independent of k≥1. Writing g=glo−dellℋ, we obtain the sharp comparison g≤glo−dellB≤ g+1 and findimBop≤ g+1. Thus the two global levels are finite simultaneously. For wide hearts the relative levels equal their classical counterparts. We also characterize syzygy finiteness of B by a finite relative syzygy stratum and obtain explicit bounds for both finitistic dimensions. If the heart is Frobenius, it is exactly the category of finitely generated Gorenstein-projective B-modules under this realization, and both self-injective dimensions of B are at most one. We also give examples to show the main results.
Related
- Relative derived delooping levels and torsion intervals
- Relative Derived Delooping Levels and τ-Tilting Modules
- Relative Suspension and Higher Delooping for Support τ-Tilting Modules
- Derived Delooping Levels of One-Point Extensions and Finitistic Dimensions
- Reduction techniques for the derived delooping levels
BibTeX
@misc{cao2026relative,
title = {{Relative delooping levels on intervals of torsion classes}},
author = {Weiqing Cao and Mingfei Xu and Xiaojin Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33762},
eprint = {2610.33762},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33762},
}
RIS
TY - UNPB
TI - Relative delooping levels on intervals of torsion classes
AU - Weiqing Cao
AU - Mingfei Xu
AU - Xiaojin Zhang
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33762
UR - https://arxiv.org/abs/2610.33762
AB - Let U be a support τ-tilting module over a finite-dimensional algebra A, and let 𝒱⊆FacU be any torsion class. The exact heart ℋ=FacU∩𝒱^⊥ has an Ext-progenerator P. We prove that Hom_A(P,−) realizes ℋ as a submodule-closed exact subcategory of modB, where B=End_A(P), and preserves minimal syzygies. The global relative k-delooping levels are independent of k≥1. Writing g=glo−dellℋ, we obtain the sharp comparison g≤glo−dellB≤ g+1 and findimB^op≤ g+1. Thus the two global levels are finite simultaneously. For wide hearts the relative levels equal their classical counterparts. We also characterize syzygy finiteness of B by a finite relative syzygy stratum and obtain explicit bounds for both finitistic dimensions. If the heart is Frobenius, it is exactly the category of finitely generated Gorenstein-projective B-modules under this realization, and both self-injective dimensions of B are at most one. We also give examples to show the main results.
ER -