Introduction

This lecture course is a continuation of the course Algebraic Geometry 1 which covered the definition of schemes, and some basic notions about schemes and scheme morphisms: Reduced and integral schemes, immersions and subschemes, the functorial point of view, fiber products of schemes, separated and proper morphisms.

The main object of study of this term’s course will be the notion of ${\mathscr O}_X$-module, a natural analogue of the notion of module over a ring in the context of sheaves of rings. As we will see, the ${\mathscr O}_X$-modules on a scheme $X$ contain a lot of information about the geometry of this scheme, and we will study them using a variety of methods. In the second part, we will introduce the notion of cohomology groups, a powerful algebraic tool that makes its appearance in many areas of algebra and geometry.

These notes are not complete lecture notes (most proofs are omitted in the notes), but should rather be thought of as a rough summary of the content of the course.