Counterexamples and gap phenomena for derived delooping levels

@misc{chen2026counterexamples,
  title = {{Counterexamples and gap phenomena for derived delooping levels}},
  author = {Liang Chen},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.35849},
  eprint = {2609.35849},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.35849},
}

TY  - UNPB
TI  - Counterexamples and gap phenomena for derived delooping levels
AU  - Liang Chen
PY  - 2026
DA  - 2026/10/04
PB  - arXiv
DO  - 10.48550/arxiv.2609.35849
UR  - https://arxiv.org/abs/2609.35849
AB  - Guo and Igusa introduced the derived and sub-derived delooping levels as refinements of the delooping level and asked how these two invariants compare, whether the derived delooping level always coincides with the opposite big finitistic dimension, and, if not, how large their difference can be. We answer these questions. First, we prove that every Artin algebra A satisfies ddell A≤ sub−ddell A. This gives a positive answer to the ordinary-degree comparison problem, whereas a single explicit eight-dimensional algebra C satisfies k−sub−ddell C=0<1=k−ddell C for k≥ 2. We then compute, for every k≥ 1, all k-derived and k-sub-derived delooping levels of the monomial family of Barrios--Lanzilotta--Mata. In particular, that family satisfies k−ddell A_n,s=Findim(A_n,s^op) for k≥ 1. In ordinary degree, this recovers an equality previously announced by Lam. We also construct a two-vertex, eleven-dimensional monomial algebra Λ, over an arbitrary field, such that Findim(Λ^op)=1<2=ddellΛ=dellΛ. The same strict inequality is obtained by Gao, Liu and Xu using different algebras. A ten-dimensional non-monomial variant with a commutative corner algebra is also given. Finally, we construct a family of finite-dimensional monomial algebras B_r satisfying Findim(B_r^op)=1 and ddell B_r=2r+1. Thus the difference ddell A−Findim(A^op) is unbounded even among monomial algebras with fixed opposite big finitistic dimension.
ER  -