# Serre dimension and stability conditions

## TL;DR

This paper explores the relationship between Serre dimension and stability conditions in triangulated categories, establishing inequalities, characterizations, and classifications related to Calabi-Yau categories and Gepner stability conditions.

## Contribution

It introduces a fundamental inequality linking Serre dimension and global dimension, characterizes Gepner type stability conditions via Serre dimension, and classifies categories with low Serre dimension.

## Key findings

- Proved an inequality between Serre dimension and global dimension.
- Characterized Gepner type stability conditions using Serre dimension.
- Classified categories with Serre dimension less than one that admit Gepner stability conditions.

## Abstract

We study relations between the Serre dimension defined as the growth of entropy of the Serre functor and the global dimension of Bridgeland stability conditions due to Ikeda-Qiu. A fundamental inequality between the Serre dimension and the infimum of the global dimensions is proved. Moreover, we characterize Gepner type stability conditions on fractional Calabi-Yau categories via the Serre dimension, and classify triangulated categories of the Serre dimension lower than one with a Gepner type stability condition.