Introduction

We describe some basic properties of the derived category of coherent sheaves on a variety (bounded derived category or perfect complexes).

The first chapter considers the problem of extending vector bundles from an open subset. Thomason and Trobaugh provided an answer to this problem by considering extensions for perfect complexes. This has applications to higher K-theory.

In the second chapter, we explain how to characterize subcategories corresponding to objects supported by a given closed subvariety. This permits a reconstruction of the variety (viewed as a ringed space) from a categorical structure. In the case of derived categories, this requires also the tensor structure.

We start with the classical case of the category of coherent sheaves (after Gabriel). We present afterwards a similar approach in the triangulated case, where serious difficulties arise.

Finally, we explain how to deduce that a smooth projective variety with ample or anti-ample canonical bundle is determined by its derived category.

We haven't included proofs of the results on general properties of abelian or triangulated categories (cf [KaScha, Nee3] for proofs). The only difficult part is Lemma 3.9 on compact objects.

This text is based on lectures at the conference Géométrie algébrique complexe, CIRM, Luminy in December 2003. I thank Paul Balmer for useful comments.

Notations

We fix a field k and we call variety a separated scheme of finite type over k.