Algebraic spaces and stacks

Algebraic spaces from étale quotients, stacks and gerbes from descent and torsors, Artin's algebraicity criteria and moduli stacks, the stack of curves and stable reduction, and simplicial models for higher stacks.

Lessons

1. Algebraic spaces
2. The bootstrap theorem
3. Properties and morphisms of algebraic spaces
4. Stacks in groupoids
5. Algebraic stacks
6. Quotient stacks and Deligne–Mumford stacks
7. Artin's axioms
8. Moduli stacks are algebraic
9. The stack of curves
10. Stable reduction and properness
11. Simplicial sets, nerves and Kan complexes
12. Test categories and Grothendieck's homotopy programme

Download editable course sources

Original text CC0 1.0. Lessons 6–9 adapt proofs by the Stacks project authors, as distributed in the AI Integrated Stacks Project, and retain the GNU FDL 1.2 or any later version (licenses/GFDL-1.2.txt).