Serre functors and graded categories

TL;DR

This paper investigates Serre structures in enriched and graded categories, exploring their preservation under various constructions and linking them to graded Frobenius algebras and fractional Calabi-Yau properties.

Contribution

It introduces new insights into Serre structures in graded categories, their invariance under orbit and skew group constructions, and connects these to graded Nakayama automorphisms and Calabi-Yau conditions.

Findings

Serre structures are preserved under orbit and skew group categories.

Derived categories of d-representation finite algebras are fractionally Calabi-Yau under certain conditions.

Connections established between Serre structures, graded Frobenius algebras, and Nakayama automorphisms.

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.