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, and fiber products of schemes.

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 particular, 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, but should rather be thought of as a rough summary of the content of the course. The notes originate from a similar course which I taught in 2019, and I will make some additions and changes (also concerning the selection of material treated in the course) as the term progresses.

References. The books [ GW1 ] by Wedhorn and myself, by Hartshorne [ H ] , and by Mumford [ Mu ] . More precise references are given in most of the individual sections. Mumford still uses the ancient terminology and calls a prescheme what we call a scheme, and a scheme what we will call a separated scheme. Further references: [ Stacks ] , [ EGA ] .