Math 193a: Algebraic Stacks, Fall 2011

Course Description

Algebraic stacks arise naturally as solutions to classification (moduli) problems, so it is desirable to understand their geometry. In this course, we will assume a working knowledge of the geometry of schemes. We will extend the definitions and techniques used to study schemes to algebraic spaces and algebraic stacks. Topics will include Grothendieck topologies, descent, algebraic spaces, fibered categories, and algebraic stacks.

Specific topics will be included based on feedback from students.

Resources

The stacks project by Johan de Jong et. al.
Algebraic spaces by Donald Knutson
Champs algébriques by Gérard Laumon and Laurent Moret-Bailly
my notes from Martin Olsson's course at Berkeley (source available in an svn repo)
Algebraization of Formal Moduli I by Michael Artin

Possible Topics

• Grothendieck topologies (using sieves?), topoi
  • Descent for sheaves in a site
  • bootstrapping properties of objects/morphisms
• Algebraic spaces
  • affine/(finite etale) = affine
  • Artin's results on modifications of algebraic spaces
• Torsors and $H^1$, gerbes and $H^2$
• Fibered categories
  • splittings
  • 2-Yoneda lemma
• Algebraic stacks
  • criteria for representability
  • DM $\Leftrightarrow$ unramified diagonal
  • coarse/good moduli space morphisms, Keel-Mori
  • DM stacks etale locally quotients by stabilizer
  • ZMT, Chow
  • valuative criteria
  • Artin representability