On the 21st of June 2021, I defended my BSc thesis in Triangulated Categories. The thesis can be found here or you can read the abstract below.

Abstract

In this paper, we give a short introduction to the theory of triangulated categories. We present the relevant definitions and properties of triangulated categories which we use to investigate the Verdier localisation of triangulated categories. Furthermore, we show that the stable category of a Frobenius category is triangulated. By exhibiting the category of chain complexes as a Frobenius category we may take its stable category, which turns out to coincide with the homotopy category, thus showing that the homotopy category is triangulated. Then, by Verdier localising the homotopy category with respect to the subcategory of acyclic complexes (resp. \(\mathcal{X}\)-acyclic complexes), we obtain the derived category (resp. \(\mathcal{X}\)-relative derived category), proving it to be triangulated. Finally, using the theory of triangulated categories, we prove that the (Gorenstein) derived category of an abelian category A is abelian if and only if A is semisimple.