Derived categories of coherent sheaves

TCC course. Thursdays 10:00 to 12:00 starting on October 17.
Lecturer: Clemens Koppensteiner, clemens.koppensteiner@maths.ox.ac.uk
No class on November 14 and December 5, but an additional class on December 12.

Synopsis

In recent years much progress has been made in understanding derived categories of coherent sheaves. These categories are interesting because they encode much of the geometry of the underlying scheme. They are also useful tools to encode and study data coming from other fields, such as the representation theroy of algebraic groups.

In the first part of the course we will introduce the main tools for the study of categories of coherent sheaves. It will in particular contain a detailed discussion of Grothendieck (coherent) duality, a far reaching extension of Serre duality. In the second part, we will discuss various applications and structure theory. Depending on time and interest of the participants, these could for example include the Bondal--Orlov reconstruction theorem, semi-orthogonal decompositions and homological projective duality, and Bridgeland's perverse coherent sheaves.

Prerequisites: Schemes and basics of coherent sheaves; a basic understanding of homological algebra and, ideally, derived categories (though we will give a short review of these).

Notes

Version of November 28

References

Many additional references can be found in the notes.

Outline

Date Topics
Oct 17 Derived categories; derived category of coherent sheaves; standard functors
Oct 24 perfect complexes, compact and dualizable objects; projection formula; Fourier-Mukai functors
Oct 31 Grothendieck duality: adjoints, dualizing complexes, Serre duality
Nov 7 Lemma on way-out functors; Bondal-Orlov reconstruction theorem
Nov 21 Bondal-Orlov reconstruction theorem; Beilinson's exceptional collection; tilting
Nov 28 Semi-orthogonal decompositions; Koszul duality
Dec 12 (Singular support; further topics)