Support τ-tilting posets and Hochschild reconstruction for matrix centralizer algebras
Abstract
Let R be a field and A the endomorphism algebra of a finite direct sum of cyclic modules over a finite-dimensional commutative local principal ideal R-algebra. We construct a central quotient showing that the support τ-tilting poset of A is isomorphic to the poset of the symmetric group with the weak order. We show that the center of A and degree-zero Hochschild homology, viewed as a module over the center, determine the truncated local algebra and the multiset of successive length gaps. For centralizer matrix algebras, the support τ-tilting poset determines the multiset of distinct-exponent counts of the primary blocks. The corresponding algebra--module pair also recovers their local algebras and gap multisets. Combining this reconstruction with the known derived equivalence classification, we characterize derived equivalence by isomorphism of these algebra-module pairs. We apply the results to Morita reconstruction in the string and gentle classes.
Related
- Categorical reconstruction of del Pezzo surfaces: Hochschild--Serre algebras and spinor modifications
- Support τ-tilting modules over Morita context algebras: A bilateral approximation approach
- BV structure on the Hochschild cohomology of Liu-Schulz algebras in characteristic two
- Hochschild theory of multiplicative sequences of algebras and coalgebra measurings
- Group action of Hochschild-Serre algebra and categorical reconstruction
BibTeX
@misc{hu2026support,
title = {{Support $τ$-tilting posets and Hochschild reconstruction for matrix centralizer algebras}},
author = {Jiangsheng Hu and Yu-Zhe Liu and Tiwei Zhao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.15460},
eprint = {2608.15460},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.15460},
}
RIS
TY - UNPB
TI - Support τ-tilting posets and Hochschild reconstruction for matrix centralizer algebras
AU - Jiangsheng Hu
AU - Yu-Zhe Liu
AU - Tiwei Zhao
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2608.15460
UR - https://arxiv.org/abs/2608.15460
AB - Let R be a field and A the endomorphism algebra of a finite direct sum of cyclic modules over a finite-dimensional commutative local principal ideal R-algebra. We construct a central quotient showing that the support τ-tilting poset of A is isomorphic to the poset of the symmetric group with the weak order. We show that the center of A and degree-zero Hochschild homology, viewed as a module over the center, determine the truncated local algebra and the multiset of successive length gaps. For centralizer matrix algebras, the support τ-tilting poset determines the multiset of distinct-exponent counts of the primary blocks. The corresponding algebra--module pair also recovers their local algebras and gap multisets. Combining this reconstruction with the known derived equivalence classification, we characterize derived equivalence by isomorphism of these algebra-module pairs. We apply the results to Morita reconstruction in the string and gentle classes.
ER -