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PreprintCC BY 4.0arXiv

Compatible gradings on dg models and cluster equivalence

  • Shengyong Pan
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Abstract

Let La1 and La2 be finite-dimensional ta2-finite complex algebras of global dimension at most two. We prove that, if they are cluster equivalent, then they admit bbZ-gradings for which the bounded derived categories of graded modules are triangle equivalent. The equivalence is not required to commute with the internal grading shifts. The proof uses common multigraded dg models, relative gradings on two-term silting objects, and covering quotients. We also consider tilting lifts of cluster-tilting objects with endomorphism algebras of global dimension at most two. In acyclic cluster type, the existence of such a lift for every cluster-tilting object implies piecewise heredity. We construct a counterexample without this assumption using a surface algebra associated with a one-holed torus.

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Cite

BibTeXRIS
BibTeX
@misc{pan2026compatible,
  title = {{Compatible gradings on dg models and cluster equivalence}},
  author = {Shengyong Pan},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.04328},
  eprint = {2611.04328},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.04328},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Compatible gradings on dg models and cluster equivalence
AU  - Shengyong Pan
PY  - 2026
DA  - 2026/11/03
PB  - arXiv
DO  - 10.48550/arxiv.2611.04328
UR  - https://arxiv.org/abs/2611.04328
AB  - Let La_1 and La_2 be finite-dimensional ta_2-finite complex algebras of global dimension at most two. We prove that, if they are cluster equivalent, then they admit bbZ-gradings for which the bounded derived categories of graded modules are triangle equivalent. The equivalence is not required to commute with the internal grading shifts. The proof uses common multigraded dg models, relative gradings on two-term silting objects, and covering quotients. We also consider tilting lifts of cluster-tilting objects with endomorphism algebras of global dimension at most two. In acyclic cluster type, the existence of such a lift for every cluster-tilting object implies piecewise heredity. We construct a counterexample without this assumption using a surface algebra associated with a one-holed torus.
ER  -