Lectures on dg-categories, Apr 13-16, 2010
Bertrand Toen (U. Montpellier)
Lecture 1: Tue 13 April 15:30-17:00
Lecture 2: Wed 14 April 15:30-17:00
Lecture 3: Thu 15 April 15:30-17:00
Lecture 4: Fri 16 April 13:30-15:00, 15:30-17:00
Lecture 1: Generalities on dg-categories.
Lecture 2: Moduli 1: moduli space of simple objects.
Lecture 3: Moduli 2: moduli of non simple objects and higher stacks.
Lecture 4: Topological and motivic invariants of dg-categories.
In this first lecture I will introduce the notions of dg-categories and of derived Morita equivalences. We explain how these two notions organize into a symmetric monoidal 2-category, and make the link with the 2-category of triangulated categories. Some examples of results concerning derived categories of algebraic varietes and schemes are given using this 2-categorical setting.
We discuss the general problem of constructing an algebraic moduli of objects in derived categories. We will present a first solution to this problem by stating the existence of an algebraic space of compact and simple objects in a nice enough dg-category. We provide applications of the existence a such a moduli space in the study of derived category of algebraic varieties. In a second part, we study the existence of a moduli space for non necessarily simple objects, and state the existence of an algebraic (higher) stack classifying all compact objects in a nice dg-category. As an application we present the construction of Hall algebras in the derived setting by means of geometric methods.
In this last lecture we introduce a topological K-theory of a dg-category, constructed using the existence of the moduli stack of compact objects presented in the previous lectures. We state several conjectures about it, and explain its motivic origin. We discuss the consequences of these conjectures on the theory of "non-commutative motives" and of "non-commutative Hodge structures".