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Equivariant Cleft Extensions and Singular Equivalences

  • Miltiadis Karakikes
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Abstract

We study the equivariant lifting of cleft extensions of abelian categories and its impact on singularity categories. Specifically, we establish the necessary framework for lifting a cleft extension to a G-equivariant cleft extension. Furthermore, we prove that a restriction functor associated to a cleft extension induces a singular equivalence if and only if its equivariant counterpart does. As a concrete application, we demonstrate that the skew group ring of a G-equivariant θ-extension is isomorphic to a θ̂-extension of the base skew group ring, allowing us to lift singular equivalences for these structures.

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Cite

BibTeXRIS
BibTeX
@misc{karakikes2026equivariant,
  title = {{Equivariant Cleft Extensions and Singular Equivalences}},
  author = {Miltiadis Karakikes},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2606.25560},
  eprint = {2606.25560},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.25560},
}
RIS
TY  - UNPB
TI  - Equivariant Cleft Extensions and Singular Equivalences
AU  - Miltiadis Karakikes
PY  - 2026
DA  - 2026/10/31
PB  - arXiv
DO  - 10.48550/arxiv.2606.25560
UR  - https://arxiv.org/abs/2606.25560
AB  - We study the equivariant lifting of cleft extensions of abelian categories and its impact on singularity categories. Specifically, we establish the necessary framework for lifting a cleft extension to a G-equivariant cleft extension. Furthermore, we prove that a restriction functor associated to a cleft extension induces a singular equivalence if and only if its equivariant counterpart does. As a concrete application, we demonstrate that the skew group ring of a G-equivariant θ-extension is isomorphic to a θ̂-extension of the base skew group ring, allowing us to lift singular equivalences for these structures.
ER  -