# 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 December 27, 2019
### References

- D. Huybrechts, Fourier-Mukai transforms in algebraic geometry. Oxford University Press, 2006.
- Amnon Neeman, Grothendieck duality made simple. ArXiv: 1806.03293
- Alexei Bondal, Dmitri Orlov, Derived categories of coherent sheaves. Proceedings of the ICM, Vol. II (Beijing, 2002), 47--56. ArXiv: math/0206295

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; introduction to categorification |