Introduction to Schemes

Instructors:	Daniele Agostini (Lectures), Clemens Nollau (Exercises).
Lectures:	Di. 14:15-15:45. Lecture Room N14 (C-Building Morgenstelle) Mi. 14:15-15:45. Lecture Room N15 (C-Building Morgenstelle).
Exercises:	TBA
Language:	English.
Registration:	URM.

Announcements

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Course Description

The course will be an introduction to Grothendieck's schemes, which give the basic language of modern algebraic geometry. We will aim to cover schemes, sheaves and cohomology.

Prerequisites

The language of schemes is fairly technical, and the geometric meaning is not immediately clear. That's why previous knowledge of the the classical language of algebraic varieties is very useful. One should also know some basic commutative algebra and some basic facts on sheaves.

The minimum prerequisites are in

H. Markwig, Introduction to Commutative Algebra and Algebraic Geometry. Online notes.

It would be even better to know the material from

A. Gathmann, Algebraic Geometry. Online notes. Chapters 1-4.

The best background could be acquired from

D. Mumford, The Red Book of Varieties and Schemes. Springer. Chapter 1.
R. Hartshorne, Algebraic Geometry. Springer. Chapter I.

References

We will not follow exactly any particular book, but the main inspirations for the course will be

A. Gathmann, Algebraic Geometry. Online notes. (Gathmann has different versions of these notes on his website, we will follow those from 2002/03).
R. Vakil, The Rising Sea: Foundations of Algebraic Geometry. Princeton University Press. There is also a free version.
R. Hartshorne, Algebraic Geometry. Springer.
D. Mumford, The Red Book of Varieties and Schemes. Springer.

There are many more useful books on this topic. Some of them are:

Q. Liu, Algebraic Geometry and Arithmetic Curves. Oxford University Press.
U. Görtz and T. Wedhorn, Algebraic Geometry I: Schemes. Springer.
D. Eisenbud and J. Harris, The Geometry of Schemes. Springer.
G. Ellingsrud and J. Ottem, Introduction to Schemes. Online notes.

The definitive (and technical) references are:

A. Grothendieck, Éléments de géométrie algébrique (EGA). Publications Mathématiques de l'IHÉS.
The Stacks Project authors, The Stacks project. Online project.

Course Log

A brief summary of each lecture, together with references to the notes, will appear below.

Exam

We are going to have oral exams. There are no formal requirements to access the exam, but students are strongly encouraged to try to solve the exercise sheets and to take active part in the lectures and the exercise sessions.

Exercise Sheets

The exercise sheets will appear below. If you want your exercise sheets to be corrected, you should upload them to URM by the deadline.