A note on non-unique enhancements

TL;DR

This paper provides a simple example of a triangulated category over a field that admits two distinct enhancements, addressing a question about the uniqueness of such enhancements.

Contribution

It presents the first explicit example of a triangulated category with multiple non-unique enhancements over a field.

Findings

Demonstrates non-uniqueness of enhancements in a specific triangulated category

Answers a previously open question by Canonaco and Stellari

Provides insight into the structure of enhancements in triangulated categories

Abstract

We give an easy example of a triangulated category, linear over a field $k$, with two different enhancements, linear over $k$, answering a question of Canonaco and Stellari.