Main Page/Reading Seminar Stacks (2025): Difference between revisions
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| | |02/28/2025 | ||
|Hairuo | |Hairuo | ||
|Grothendieck Topologies / Sites | |Grothendieck Topologies / Sites | ||
|Introduction to Grothendieck Toplogies /sites. More information can be found in [https://www.math.nagoya-u.ac.jp/~larsh/teaching/S2013_AG/grothendiecktopologies.pdf Notes on a Seminar by Michael Artin]. If one wishes to presert more on the \'etale site, [https://www.jmilne.org/math/CourseNotes/LEC.pdf Milne's Lecture Notes] has far more details. | |Introduction to Grothendieck Toplogies /sites. More information can be found in [https://www.math.nagoya-u.ac.jp/~larsh/teaching/S2013_AG/grothendiecktopologies.pdf Notes on a Seminar by Michael Artin]. If one wishes to presert more on the \'etale site, [https://www.jmilne.org/math/CourseNotes/LEC.pdf Milne's Lecture Notes] has far more details. | ||
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| | |03/07/2025 | ||
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|Fibred Categories | |Fibred Categories | ||
|Introduction to fibred categories. Describe correspondence between fibred categories /C and presheaves on C. Groupoids in C, fibre products of fibred categories, Yoneda Lemma, and discussion of categories fibred in groupoids. Discuss examples. The following exercises in Olsson are relevant for the future: 3.A, 3.B, 3.C, 3.D, 3.F, and 3.G. | |Introduction to fibred categories. Describe correspondence between fibred categories /C and presheaves on C. Groupoids in C, fibre products of fibred categories, Yoneda Lemma, and discussion of categories fibred in groupoids. Discuss examples. The following exercises in Olsson are relevant for the future: 3.A, 3.B, 3.C, 3.D, 3.F, and 3.G. | ||
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| | |03/14/2025 | ||
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|Descent and Stack Conditions | |Descent and Stack Conditions | ||
| | |Discuss generalities on descent. Explain why fppf descent and fpqc descent work. Applications of descent e.g. closed subschemes, open embeddings, affine morphisms, polarized schemes. Discuss torsors and principal homogeneous spaces. See exercise 4.G for invertible sheaves on sites. | ||
Definition of stack using fibred categories. Fibred products of stacks, explanation of the stackification functor. | |||
Examples of stacks: Olsson's examples are found in the exercises 4.C, 4.E, 4.H. If exercises are too hard, see Laumon-Moret-Bailly. | |||
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|Algebraic Spaces Part 1 | |Algebraic Spaces Part 1 | ||
| | |TBD. Olsson's presentation spans three chapters. Alper's notes are really good for this. Note the Stacks project uses the fppf topology instead of the etale topology. | ||
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|Algebraic Spaces Part 2 | |Algebraic Spaces Part 2 | ||
| | |TBD | ||
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|Algebraic Spaces Part 3 | |Algebraic Spaces Part 3 | ||
| | |TBD | ||
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Revision as of 01:51, 25 February 2025
Logistics: TBD
Tentative schedule
| date | speaker | title | topics |
|---|---|---|---|
| 02/28/2025 | Hairuo | Grothendieck Topologies / Sites | Introduction to Grothendieck Toplogies /sites. More information can be found in Notes on a Seminar by Michael Artin. If one wishes to presert more on the \'etale site, Milne's Lecture Notes has far more details. |
| 03/07/2025 | Fibred Categories | Introduction to fibred categories. Describe correspondence between fibred categories /C and presheaves on C. Groupoids in C, fibre products of fibred categories, Yoneda Lemma, and discussion of categories fibred in groupoids. Discuss examples. The following exercises in Olsson are relevant for the future: 3.A, 3.B, 3.C, 3.D, 3.F, and 3.G. | |
| 03/14/2025 | Descent and Stack Conditions | Discuss generalities on descent. Explain why fppf descent and fpqc descent work. Applications of descent e.g. closed subschemes, open embeddings, affine morphisms, polarized schemes. Discuss torsors and principal homogeneous spaces. See exercise 4.G for invertible sheaves on sites.
Definition of stack using fibred categories. Fibred products of stacks, explanation of the stackification functor. Examples of stacks: Olsson's examples are found in the exercises 4.C, 4.E, 4.H. If exercises are too hard, see Laumon-Moret-Bailly. | |
| Algebraic Spaces Part 1 | TBD. Olsson's presentation spans three chapters. Alper's notes are really good for this. Note the Stacks project uses the fppf topology instead of the etale topology. | ||
| Algebraic Spaces Part 2 | TBD | ||
| Algebraic Spaces Part 3 | TBD | ||
| Algebraic Stacks | |||
| Expedition: Quot Scheme and Hilbert Scheme | |||
| Quasicoherent Sheaves on Algebraic Stacks | |||
| Coarse Moduli Spaces and Moduli of Curves | |||
| Gerbes | |||
| Local Structure of Algebraic Stacks | |||
| Coarse Moduli Spaces and Geometric Invariant Theory | |||
| Future Potential Topics | Bun_G (perhaps following Sorger), Artin Algebraization, Formal Moduli, etc. We'll figure it out when we get there. |
References
- Martin Olsson's Algebraic Spaces and Stacks
- Laumon-Moret-Bailly Champs Algebriques
- Alper's Stacks and Moduli