Geometria degli schemi

The first part (6 cfu) aims at giving an introduction to the theory of schemes. A scheme is a vast algebraic generalization of the concept of topological variety and allows to deal with objects which are apparently very different. For example, the affine line over the complex numbers or (the spectrum) of the ring of integers Z are very similar from the point of view of schemes. We will introduce the notions of scheme, of sheaf on a scheme and of morphism of schemes with plenty of examples. We will then study the cohomology of a sheaf on a scheme and its main properties.

We will try to be as much as possible self-contained. In particular, we will recall the basic definitions and results from commutative algebra which are needed. In particular, the students are not required to attend a course about commutative algebra before. Nevertheless, it could be a good idea to attend to course on commutative algebra during the first semester of the first year of the Laurea Magistrale and the course on the geometry of schemes during the fist semester of the second year.