5 - Derived categories and tilting  pp. 49-104

Derived categories and tilting

By Bernhard Keller

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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 equivalence 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 categories having equivalent derived categories. Finally, we briefly sketch a generalization of the tilting setup to differential graded algebras.


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 K0(A), its center Z(A), its higher K-groups Ki(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 Mn A, and, more generally, that it is preserved under Morita equivalence.