Teaching in summer semester 2024:
Cohomology of sheaves and simplicial homotopy:
This is an introduction to the notion of Grothendieck topologies, mostly Zariski, Nisnevich and Etale topologies, and the associated notions of sheaves of sets, cohomology theory of sheaves of abelian groups, and also probably a bit of homotopy theory (of simplicial sheaves of sets). Some applications will be given.
This lecture can be considered as a direct sequel to Algebraic Geometry 1 and 2. We will start with a quick recollection concerning the basic notions on schemes and morphisms of schemes, and then we will study the notions of smooth and E'tale morphisms. Then we will introduce the abstract notion of Grothendieck topology on a site (mostly consisting of schemes) and the corresponding notions of sheaves of sets. We will give some general porperties of these, and will give our basic examples : the Zariski, the Nisnevich and the E'tale topology on schemes. We will emphasize the role played by local rings, Henselian local rings, and respectively strictly Henselian local rings in the Zariski, Nisnevich, and E'tale topology. Then we will study the category of sheaves of abelian groups (on a reasonable site...) and prove it is a Grothendieck abelian category. We will then define the cohomology of schemes with coefficients in a sheaf of abelian groups and study their basic properties, and give basic examples. We will try to give a list of fundamental theorems, and give a sketch of proof of some of these. Depending on the remaining time, we will explain the Verdier formula which computes cohomology of sheaves of abelian groups in terms of hypercoverings of schemes, a notion involving simplicial schemes. Then slowly we will introduce the category of simplicial sheaves of sets and its associated homotopy theory, and give examples. The background will be something like Algebraic Geometry 1 and 2 and also some basics in homotopy theory.
Simplicial objects and homological algebra :
This is an introduction to the notion of chain complexes in abelian categories, leading to the concept of homological algebra, and the notion of simplicial objects in reasonable categories, leading to the notion of simplicial homotopy theory. The point of the lecture is to connect these two theories using the pair of adjoint functors consisting of the chain complex associated to a simplical object and the simplicial object associated to a non negatively graded chain complex. This adjunction also passes to a pair of adjoint functors on the homotopy categories level. This lecture can be taken in parallel to the above lecture Cohomology of sheaves and simplicial homotopy, but it is also independent (and more elementary).
Previous semesters