Arithmetic Algebraic Geometry II
The purpose of the course is to give an introduction to the theory of
sheaves and sheaf cohomology with a focus on applications in
arithmetic geometry. Some background knowledge of schemes will be
assumed. The canonical reference for this material is part 4 of the
Grothendieck school's Séminare de Géométrie
Algébrique (SGA). It is the intention that, at the end of the
course, participants should have an understanding of Grothendieck's
relative point of view on cohomology; base-change theorems and their
usage; and, if time permits, recollement
and descente.
Text: The main text is SGA 4, which is available online by
following the links below. It is a good idea to look
through the lists of contents to get a sense of the distribution of
the material.
A useful shorter text is
Artin's Grothendieck
Topologies. Another comprehensive source is
the stacks
project book.
Lectures: Tuesdays 1:15 p.m. - 3:00 p.m. in A103 and Fridays
9:15 a.m. - 10:00 a.m. in Auditorium
10.
- Lecture 1: Sheaves.
- Lecture 2: Sheafification.
- Lecture 3: The functoriality of categories of sheaves.
- Lecture 4: Abelian categories and derived functors.
- Lecture 5: Grothendieck's small object argument.
- Lecture 6: Cohomology.
- Lecture 7: The étale topology.
- Lecture 8: Faithfully flat descent.
- Lecture 9: The étale topos.
- Lecture 10: The Lefschetz trace formula.
Updated Lecture Notes
and handwritten notes for Lecture 10.
Recitations: Mondays 11:15 a.m. - 12:00 noon and 1:15 p.m - 3:00 p.m. in Room 4.01 with Kristian Moi as instructor.
Problems for recitations: