A compactly generated triangulated category

Specialeforsvar ved Rikke Stuart Christensen

Titel: A compactly generated triangulated category

Abstract: The goal of this thesis is to prove that the homotopy category of complexes of projective modules, $\mathcal{K}(\Pro \A)$ is a compactly generated triangulated category over a coherent ring, which we throughout the thesis will refer to as \emph{the main theorem}. To do this, we begin with the definition of a projective module. We dedicate chapter one to this, along with some properties for projective modules, that we will use throughout the thesis. This should be well-known, but is necessary in order to establish definitions, nota-tions and concepts used throughout the thesis. We then define homotopy and the homotopy category of complexes in the beginning of chapter two. Our first milestone is to proof that the homotopy category is a triangulated category, so we dedicate the rest of chapter two to triangulated categories. In order to get a better understanding of triangu-lated categories, we compare them to abelian categories. It turns out that a triangulated category is not necessarily abelian. We will get more into that in the chapter. Next we introduce the total Hom complex, which is Hom taken over complexes, and some proper-ties for this. Chapter three also gives a construction that we will need in the main theorem to describe the compact objects in $\mathcal{K}(\Pro R)$, the homotopy category of projective modules, which will mark the end of chapter three. In the fourth chapter we are ready to prove the main theorem. To do this we introduce the equational criterion; a tool to see if a module is flat. It brings us on a minor detour, using a whole section for this criterion. Furthermore, to prove this criterion, we need to introduce Tor, which we will do in the beginning of this chapter. We finish the chapter with the proof of the main theorem. In the fifth and final chapter we discuss the connection between short exact sequences and distinguished triangles. Furthermore we introduce the derived category obtained by localizing over the quasi-isomorphisms in the homotopy category since there is a connection between this and the compact objects in the homotopy category. This is the main result of chapter five and with that we will finish the chapter and thereby the thesis

Vejleder: Henrik G. Holm
Censor:   Henning Haahr Andersen, Aarhus Universitet