UE An introduction to algebraic geometry

Diplômes intégrant cet élément pédagogique :


Algebraic geometry is roughly the study of objects obtained as solutions of system of polynomials, combining geometric intuition and algebraic methods. It is one of the richest and most interesting field of mathematics, having connections with diverse branches such as topology and number theory. Its successes are manifold, including among other the proof of Fermat s conjecture by Wiles or the proof of Weil's conjectures by Deligne and Grothendieck.

The goal of this course is to provide solid ground for the more advanced notions in algebraic geometry. In the first part, we will focus on affine schemes. We will introduce the Zariski topology and study its behaviour under ring homomorphisms, focusing on localizations and quotient by ideals. We will then briefly discuss the notion of sheaf on a topological space before turning to the notions of scheme and morphism of schemes. In the second part, we will study more specific properties of morphisms and useful constructions such as the fiber product. We will also spend a fair amount of time on the notion of point. Time permitting, we will introduce the Chow groups associated to a scheme.

Informations complémentaires

Langue(s) : Anglais