Strong uniqueness of enhancements for the dual numbers: a case study

TL;DR

This paper proves the strong uniqueness of dg enhancements for derived categories over dual numbers and hereditary categories, providing a complete classification of indecomposables and relating categories to sequences of vector spaces.

Contribution

It establishes the strong uniqueness of dg enhancements for derived categories over dual numbers and hereditary categories, with a new classification approach.

Findings

Derived categories over dual numbers have strongly unique dg enhancements.

All derived categories of hereditary categories have strongly unique enhancements.

Complete classification of indecomposable objects achieved.

Abstract

We prove that the bounded and bounded below derived categories of (all) modules over the dual numbers have strongly unique (dg) enhancements. To this end we relate those categories to the category of sequences of vector spaces, which allows a complete classification of indecomposable objects. Along the way we also prove that all the derived categories of any hereditary category have strongly unique enhancements.