Lecturer:
Course Type:
PhD Course
Academic Year:
2015-2016
Period:
From Jan 11
Duration:
40 h
Description:
Webpage: https://algebraicstackssissa2016.wordpress.com/
Content:
- Schemes as functors; (pre)sheaves as functors. Algebraic spaces. There is no moduli space for genus g curves.
- Grothendieck topologies: Zariski and étale topology. Schemes as sheaves, descent.
- Groupoids as a 2-category: 2-commutative diagrams, 2-fibered product, 2-cartesian diagram, inertia groupoid. Rigid groupoids.
- Prestacks as a 2-category. Fibered categories. Stacks. Quotient stacks, moduli stacks. Stacks as 2-category. Inertia stack.
- Schemes and algebraic spaces as stacks. Representable and strongly representable morphisms. Algebraic stacks in the sense of Deligne-Mumford and Artin.
- Closed and open substacks. Properness and separatedness for morphisms of algebraic stacks, valuative criteria.
- Infinitesimal study of algebraic stacks, smoothness and étaleness, cotangent complex.
- Coherent sheaves on algebraic stacks. If time allows, Riemann-Roch theorem for smooth Deligne-Mumford stacks.
An significant part of the course will be devoted to explicit examples.
Note: 20 h Jan/Feb + 20 h Apr/May
Prerequisites:
A working knowledge of schemes, including sections 2.1-2.8 of Hartshorne, and classical algebraic geometry for the examples.
Research Group:
Location:
A-134