Abstract
We study Serre structures on categories enriched in pivotal monoidal categories, and apply this to study Serre structures on two types of graded k-linear categories: categories with group actions and categories with graded hom spaces. We check that Serre structures are preserved by taking orbit categories and skew group categories, and describe the relationship with graded Frobenius algebras. Using a formal version of Auslander-Reiten translations, we show that the derived category of a d-representation finite algebra is fractionally Calabi-Yau if and only if its preprojective algebra has a graded Nakayama automorphism of finite order. This connects various results in the literature and gives new examples of fractional Calabi-Yau algebras.
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Acknowledgements
Thanks to Bethany Marsh for originally directing me to the fractional Calabi-Yau property. Thanks to Alex Dugas, Martin Herschend, and Gustavo Jasso for helpful discussions and pointers to the literature. Thanks to an anonymous referee for helpful comments and corrections.
Parts of this paper were written during visits to the Institut des Hautes Études Scientifiques and the Institut Henri Poincaré in Paris. Thanks to both institutions, and the Jean-Paul Gimon Fund, for financial support and for providing a great working environment.
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Grant, J. Serre Functors and Graded Categories. Algebr Represent Theor 26, 2113–2180 (2023). https://doi.org/10.1007/s10468-022-10151-4
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DOI: https://doi.org/10.1007/s10468-022-10151-4
Keywords
- Serre functor
- Orbit category
- Enriched category
- Derived Picard group
- Fractional Calabi-Yau
- Preprojective algebra