OVERVIEW

The course will give an introduction to triangulated categories and derived categories of Abelian categories, the main goal being the study of the derived category of coherent sheaves on a smooth projective variety. The main motivation for the lectures will be to answer the following question: can non-isomorphic smooth projective varieties have equivalent derived categories? The Bondal-Orlov Theorem proves that this result hold if the varieties have (anti)ample canonical divisor, but it is false in general thanks to an example of Mukai.

AIMS AND CONTENT

LEARNING OUTCOMES

The main goal of the teaching is to introduce the notion of triangulated and of derived category, and to show the deep relations between category theory and algebraic geometry. The main objective is to give the students the knowledge about some very important and modern tools in algebraic geometry that are widely spread nowadays in research.

PREREQUISITES

The prerequisites are: basic category theory (notion of category and of functor), elements of commutative algebra (in particular, the notion of module over a ring and homology), sheaves and algebraic varieties (as been introduced in Istituzioni di Geometria Superiore 2 or Geometria Superiore 1)

TEACHING METHODS

The teaching will be of traditional type, with no separation between exercices and theory.

SYLLABUS/CONTENT

1. Review of category theory. Abelian categories and fundamental examples: moduli over a commutative ring, (quasi-)coherent sheaves on a projective variety. 

2. Triangulated categories: axioms and examples. Derived category of an Abelian category: costruction, structure of triangulated category. Bounded derived categories.

3. Functor between triangulated categories. Derived functors, Serre functors.

4. Varieties with (anti)ample canonical divisor and Bondal-Orlov Theorem.

5. Fourier-Mukai functors, examples and main results. Fourier-Mukai functors and equivalences. Mukai example.

RECOMMENDED READING/BIBLIOGRAPHY

The main source for this course will be the book of D. Huybrechts, Fourier-Mukai functors in Algebraic Geometry, Oxford University Press (2006)

TEACHERS AND EXAM BOARD

Exam Board

ARVID PEREGO (President)

VICTOR LOZOVANU

MATTEO PENEGINI (President Substitute)

LESSONS

Class schedule

The timetable for this course is available here: Portale EasyAcademy