PreprintCC BY 4.0arXiv
A negative answer to a question on tilting objects and two-term complexes
Abstract
Let M be a silting object in an idempotent complete algebraic triangulated category 𝒯. Put B=End𝒯(M), and let ℙM: pr(M)→ K[−1,0](projB) be the presentation functor associated with M. It was recently asked whether ℙM(T) must be tilting whenever T∈pr(M) is a tilting object. We answer this question in the negative by giving an explicit finite-dimensional example. Namely, for Λ=k(1xrightarrowα2xrightarrowβ3xrightarrowγ4)/(αβγ), we construct a silting object M∈ Kb(projΛ) and a tilting object T=ΣΛ∈pr(M) for which dimkHomKb(projB)(ℙM(T),Σ−1ℙM(T))=1. Thus ℙM(T) is a two-term silting complex but not a tilting complex.
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BibTeX
@misc{he2026negative,
title = {{A negative answer to a question on tilting objects and two-term complexes}},
author = {Jing He and Panyue Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02023},
eprint = {2609.02023},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02023},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A negative answer to a question on tilting objects and two-term complexes
AU - Jing He
AU - Panyue Zhou
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02023
UR - https://arxiv.org/abs/2609.02023
AB - Let M be a silting object in an idempotent complete algebraic triangulated category 𝒯. Put B=End_𝒯(M), and let ℙ_M: pr(M)→ K^[−1,0](projB) be the presentation functor associated with M. It was recently asked whether ℙ_M(T) must be tilting whenever T∈pr(M) is a tilting object. We answer this question in the negative by giving an explicit finite-dimensional example. Namely, for Λ=k(1xrightarrowα2xrightarrowβ3xrightarrowγ4)/(αβγ), we construct a silting object M∈ K^b(projΛ) and a tilting object T=ΣΛ∈pr(M) for which dim_kHom_K^b(projB)(ℙ_M(T),Σ^−1ℙ_M(T))=1. Thus ℙ_M(T) is a two-term silting complex but not a tilting complex.
ER -