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Algebraic stacks

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.
Location: 
A-134

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