Advanced Algebraic Geometry, Universiteit Leiden, Fall 2013

News

• Below is a scan of the notes of lecture 1.
• Starting September 10 there is a problem session for this course in room 401 from 14:45--15:30.
• To start reading the stacks project, follow the link to it in the references below, and then click on ``browse the project online''.

Teachers

Time and place

Course outline

The aim of this course is to state the Grothendieck Hirzebruch Riemann Roch theorem for morphisms of smooth projective varieties over a field, and prove as much of it as we can. This also means that the necessary tools will be introduced:

• schemes (and their morphisms of course, and properties thereof) (3 weeks),
• sheaves of modules, coherent, quasi-coherent, divisors and Picard group (2 weeks)
• cohomology, higher direct images, some general homological algebra, spectral sequences (2 weeks),
• Grothendieck groups of coherent sheaves (K₀(X)) and of locally free coherent sheaves (K⁰(X)) and their relation (K(X):=K⁰(X)=K₀(X)) and structures (1 week),
• intersection theory, Chow group, Chow ring A(X) (2 weeks),
• Chern ring morphism from K(X) to A(X)_Q, and the Todd class (1 week).

Once we have these tools available, the general Grothendieck Hirzebruch Riemann Roch theorem is easily stated. We follow the article by Borel and Serre (see the list of references below).

In the remaining two weeks we will try to present as much as possible of the proof given in Borel-Serre:

• first reductions,
• exactness and homotopy properties for K(X),
• proof for the projection from Pⁿ x X to X,
• proof for Y ➝X the embedding of a codimension one subvariety,
• blow up, and embeddings in higher codimension...

References

Borel-Serre

Le théorème de Riemann Roch

Hartshorne

Algebraic geometry. Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977.

de Jong et al.

Stacks project.

Lecture notes

• lecture 1 by Bas, September 3.
• lecture 3 by Bas, September 17.
• lecture 5 by Bas, October 8.
• lecture 6 by Bas, October 15.
• lecture 9 by Bas, November 12

Exam