Derived equivalence classification of twisted and generalized multifold extensions of affine type

Abstract

Let B be a piecewise hereditary algebra of affine type over an algebraically closed field. In this paper we give a derived and stable equivalence classification of twisted multifold extensions of B. The types D̃ and Ẽ are already covered by the classification for tree type. We therefore consider the remaining types Ã_p,q. We show that the nth power of the Nakayama functor on the full subcategory of a transjective component determines the twist, including its action on morphisms, up to conjugacy in the derived Picard group of a hereditary algebra. Together with the hereditary derived type and the folding number, this gives a complete invariant for derived equivalence, and also for stable triangle equivalence. We then compute these conjugacy classes in the unbalanced, balanced, bypass, and Kronecker cases, keeping the finite rotation data. Finally, by the normalization theorem of Asashiba--Kimura--Nakashima--Yoshiwaki, we obtain the same classification for generalized multifold extensions with nonzero integer jump.

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BibTeX RIS

BibTeX

@misc{asashiba2026derived,
  title = {{Derived equivalence classification of twisted and generalized multifold extensions of affine type}},
  author = {Hideto Asashiba},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.25312},
  eprint = {2610.25312},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.25312},
  note = {CC BY-NC-ND 4.0},
}

RIS

TY  - UNPB
TI  - Derived equivalence classification of twisted and generalized multifold extensions of affine type
AU  - Hideto Asashiba
PY  - 2026
DA  - 2026/10/25
PB  - arXiv
DO  - 10.48550/arxiv.2610.25312
UR  - https://arxiv.org/abs/2610.25312
AB  - Let B be a piecewise hereditary algebra of affine type over an algebraically closed field. In this paper we give a derived and stable equivalence classification of twisted multifold extensions of B. The types D̃ and Ẽ are already covered by the classification for tree type. We therefore consider the remaining types Ã_p,q. We show that the nth power of the Nakayama functor on the full subcategory of a transjective component determines the twist, including its action on morphisms, up to conjugacy in the derived Picard group of a hereditary algebra. Together with the hereditary derived type and the folding number, this gives a complete invariant for derived equivalence, and also for stable triangle equivalence. We then compute these conjugacy classes in the unbalanced, balanced, bypass, and Kronecker cases, keeping the finite rotation data. Finally, by the normalization theorem of Asashiba--Kimura--Nakashima--Yoshiwaki, we obtain the same classification for generalized multifold extensions with nonzero integer jump.
ER  -

Abstract

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