DERIVED CATEGORIES AND TILTING BERNHARD KELLER Abstract. We review the basic definitions of derived categories and derived functors. We illustrate them on simple but non trivial examples. Then we explain Happel’s theorem which states that each tilting triple yields an equiv- alence between derived categories. We establish its link with Rickard’s theorem which characterizes derived equivalent algebras. We then examine invariants under derived equivalences. Using t-structures we compare two abelian cate- gories having equivalent derived categories. Finally, we briefly sketch a gener- alization of the tilting setup to differential graded algebras. Contents 1. Introduction 1 2. Derived categories 2 3. Derived functors 10 4. Tilting and derived equivalences 12 5. Triangulated categories 16 6. Morita theory for derived categories 20 7. Comparison of t-structures, spectral sequences 23 8. Algebraic triangulated categories and dg algebras 27 References 31 1. Introduction 1.1. Motivation: Derived categories as higher invariants. Let k be a field and A a k-algebra (associative, with 1). We are especially interested in the case where A is a non commutative algebra. In order to study A, one often looks at various invariants associated with A, for example its Grothendieck group K 0 (A), its center Z (A), its higher K-groups K i (A), its Hochschild cohomology groups HH ∗ (A, A), its cyclic cohomology groups .... Of course, each isomorphism of algebras A → B induces an isomorphism in each of these invariants. More generally, for each of them, there is a fundamental theorem stating that the invariant is preserved not only under isomorphism but also under passage from A to a matrix ring M n (A), and, more generally, that it is preserved under Morita equivalence. This means that it only depends on the category Mod A of (right) A-modules so that one can say that the map taking A to any of the above invariants factors through the map which takes A to its module category: A               K 0 (A),Z (A),K i (A),HH ∗ (A, A),HC ∗ (A),... Mod A                            Date : November 2003, last modified on April 21, 2004. Key words and phrases. Derived category, Tilting theory. 1