Lectures on derived and triangulated categories

These notes are meant to provide a rapid introduction to triangulated categories. We start with the definition of an additive category and end with a glimps of tilting theory. Some exercises are included.

💡 Research Summary

The notes present a concise yet thorough introduction to triangulated and derived categories, aimed at readers who have a basic familiarity with additive categories and wish to move quickly into modern homological algebra. The exposition is organized into four main parts, each culminating in a set of exercises that reinforce the material.

The first part revisits the definition of an additive category, emphasizing the existence of finite direct sums, zero objects, and additive functors. Concrete examples such as module categories, categories of chain complexes, and categories of sheaves are discussed to illustrate the abstract notions. This groundwork prepares the reader for the language of complexes that follows.

The second part introduces chain complexes and chain maps, distinguishing between homotopy equivalences and quasi‑isomorphisms. The homotopy category K(A) of an additive category A is constructed by identifying chain homotopic maps. The shift (or suspension) functor is defined by shifting degrees of a complex, and the cone construction is presented as the fundamental tool for building exact triangles. Using cones, the authors verify the four triangulated axioms (TR1–TR4), providing explicit diagrams for the octahedral axiom and showing how rotation of triangles works in practice.

The third part moves to the derived category D(A). By formally inverting all quasi‑isomorphisms in K(A), the derived category is obtained as a Verdier localization. The notes explain why this localization preserves the triangulated structure and discuss the existence of enough projectives or injectives that allows one to compute derived functors via projective or injective resolutions. Right and left derived functors are defined, and the classic examples of the left derived tensor product ⊗^L and the right derived Hom RHom are worked out in detail, illustrating how they become exact functors between derived categories.

The final part offers a glimpse of tilting theory. A tilting object T in D^b(mod‑A) is defined by two Ext‑vanishing conditions and the property that it generates the derived category. The authors show that End_A(T) is a finite‑dimensional algebra and that the derived categories D^b(mod‑A) and D^b(mod‑End_A(T)) are triangulated equivalent via the functor RHom(T, –). Both 1‑tilting and 2‑tilting cases are treated, and the connection to t‑structures and hearts is briefly mentioned, indicating how tilting provides a bridge between representation theory and algebraic geometry.

Each chapter ends with carefully chosen exercises: proving the triangulated axioms for specific cones, constructing explicit quasi‑isomorphisms, computing derived functors in concrete settings, and identifying tilting modules for small algebras. These problems are designed to move the reader from passive understanding to active manipulation of the concepts.

Overall, the notes succeed in delivering a rapid yet solid foundation in triangulated and derived categories. By progressing from additive categories through homotopy categories, triangulated structures, derived localizations, and finally tilting equivalences, the text equips the reader with the essential tools needed for further study of advanced topics such as t‑structures, perverse sheaves, and differential graded categories. The inclusion of exercises ensures that readers can test their comprehension and develop the technical proficiency required for research in modern homological algebra and its applications.