Stacks and Algebraic Stacks
Reading Course at Rutgers University, Fall 2018. Fibered categories, stacks and the theory of descent, algebraic spaces and algebraic stacks.
- Instructor
- Danny Krashen
- Meeting Time
- TBD
- Meeting Location
- TBD
Weekly Topic List
Sheaves
- Introduction to presheaves on topological spaces (Tennison, Sections 1.1, 1.2, 1.3, 1.5(first half); Stacks Project Tag 006A Sections 6.1, 6.2, 6.3, 6.4, 6.6)
- Introduction to sheaves on a topological space (Tennison, Sections 2.1, 2.1; Stacks Project Tag 006A Sections 6.7, 6.8, 6.10)
- Stalks, the sheaf space, and sheafification (Tennison, Sections 1.4, 2.3; Stacks Project Tag 006A Sections 6.11, 6.12, 6.13, 6.16, 6.17, 6.18, 6.19, 6.20)
Grothendieck Topologies
- Definitions of Grothendieck topologies and sheaves on them (Tamme, Sections I.1, I.2)
- Universally effective epimorphisms and the canonical topology (Tamme, Section I.3.1)
- Examples: G-sets, discrete and profinite cases (Tamme, Sections I.3.2, I.1.3.3)
- Sheafification of presheaves on Grothendieck topologies
Overview of goals
- Examples of stacks: Topological sheaves and their gluings, bundles, modules
- The language of pseudo-functors
- The language of fibered categories
- (Some) 2-Yoneda lemma(s)
- Topological descent and topological stacks
- Grothendieck topologies, sites and topoi
- Descent and stacks
- Algebraic geometry: functors of points and the Yoneda lemma
- Algebraic geometry: etale morphisms, formally smooth morphisms
- Algebraic geometry: the etale topology
- Algebraic spaces
- Algebraic stacks