Abelian hereditary fractionally Calabi-Yau categories

TL;DR

This paper classifies abelian hereditary fractionally Calabi-Yau categories over algebraically closed fields, identifying them as categories of representations of Dynkin quivers, nilpotent cycle representations, or coherent sheaves on elliptic curves or weighted projective lines.

Contribution

It provides a complete classification of abelian hereditary fractionally Calabi-Yau categories up to derived equivalence, introducing generalized 1-spherical objects for analysis.

Findings

Classified categories include Dynkin quiver representations, nilpotent cycle representations, and coherent sheaves on elliptic curves.

Introduced generalized 1-spherical objects to analyze tubes in hereditary categories.

Established derived equivalences among these categories.

Abstract

As a generalization of a Calabi-Yau category, we will say a k-linear Hom-finite triangulated category is fractionally Calabi-Yau if it admits a Serre functor S and there is an n > 0 with S^n = [m]. An abelian category will be called fractionally Calabi-Yau is its bounded derived category is. We provide a classification up to derived equivalence of abelian hereditary fractionally Calabi-Yau categories (for algebraically closed k). They are: the category of finite dimensional representations of a Dynkin quiver, the category of finite dimensional nilpotent representations of a cycle, and the category of coherent sheaves on an elliptic curve or a weighted projective line of tubular type. To obtain this classification, we introduce generalized 1-spherical objects and use them to obtain results about tubes in hereditary categories (which are not necessarily fractionally Calabi-Yau).