Preliminary contents
- Foundations:
- Grothendieck categories
- Fibered categories, descent and stacks
- Algebraic spaces
- Algebraic stacks and Deligne–Mumford stacks
- Quasi-coherent sheaves (perhaps cohomology)
- Examples:
- Torsors, gerbes, root stacks, weighted blow-ups
- Moduli of curves, line bundles, vector bundles, sheaves etc.
- Theorems:
- Existence of quotients/coarse moduli spaces
- Artin approximation and Artin's axioms for algebraicity
- Local structure of Artin stacks
- Existence of good moduli spaces
Prerequisites
Good knowledge of scheme theory (proficient in Hartshorne Ch. II, preferably also Ch. III, at least knowledge of flat and étale morphisms, see [O, Ch. 1]). Preferably also some knowledge of Grothendieck topologies (see [O, Ch. 2.1—2.2] and [V1]) but we will review this.Examiner
David Rydh
Course start
First meeting on Friday, Jan 26, 13:15-15:00, KTH 3721.Time and place
After that, mostly Wednesdays, 15:15–17:00, KTH 3418.The lectures will also be available via Zoom with meeting ID: 647 3432 3301 (link).
Examination
The plan is that I will give all or most lectures with homework assignments as examination. It may be that part of the examination will be to give a lecture (1–2 hours).Literature
As the main text-book we will use Martin Olsson's book.- [O] M. Olsson, Algebraic stacks, 2016. (AMS Bookstore and erratum)
- [Alp] J. Alper, Stacks and moduli, Book in progress, Nov 15, 2023. (Alper's webpage)
- [AlpAA] J. Alper, Artin algebraization and quotient stacks, Lecture notes from summer school in Mainz, 2015. (arXiv:1510.07804)
- [BCEFFGK] Behrend, Conrad, Edidin, Fantechi, Fulton, Göttsche, Kresch, Algebraic stacks, in preparation, last updated 2015. (Kresch's webpage, archived from 2018)
- [Beh] K. Behrend, Introduction to algebraic stacks. In: Brambila-Paz L, Newstead P, Thomas RP, García-Prada O, eds. Moduli Spaces. London Mathematical Society Lecture Note Series. (2014), pp. 1-131. (Cambridge Core)
- [HL] D. Halpern-Leistner, Lectures notes on moduli, Jan 4, 2021. (Online lecture notes)
- [K] D. Knutson, Algebraic spaces, 1971. (Springer Link)
- [LMB] G. Laumon & L. Moret-Bailly, Champs Algébriques, 2000. (Springer Link — but KTH does not seem to have access)
- [SP] Stacks project authors, The Stacks project. (website)
- [V1] A. Vistoli, Grothendieck topologies, fibered categories and descent theory, FGA Explained, Part 1, 2005. (last version on Vistoli's web page)
- [V2] A. Vistoli, Angelo Vistoli on Gerbes, Lecture notes from summer school in Mainz, 2015. (version from David Holmes' web page)
- [A1] M. Artin, The implicit function theorem in algebraic geometry, Algebraic Geometry, Bombay 1968
- [A2] M. Artin, Algebraic Spaces, Yale 1971
- [A3] M. Artin, Algebraic approximation of structures over complete local rings, Publications mathématiques de l'IHÉS, 36 (1969), p. 23-58. (Numdam)
- [A4] M. Artin, Algebraization of formal moduli I, 1969
- [A5] M. Artin, Versal deformations and algebraic stacks, Inventiones, 1974. (SpringerLink)
- [AlpGM] J. Alper, Good moduli spaces, Ann. Inst. Fourier, 63 (2013), p. 2349-2402. (Numdam)
- [AHLH] J. Alper, D. Halpern-Leistner & J. Heinloth, Existence of moduli spaces for algebraic stacks, Inv. Math. 2023. (SpringerLink)
- [AHR] J. Alper, J. Hall & D. Rydh, A Luna étale slice theorem for algebraic stacks, Ann. Math. 2020. (ProjectEuclid)
- [DM] P. Deligne & D. Mumford, The irreducibility of the space of curves of given genus, Publications mathématiques de l'IHÉS, 36 (1969), p. 75-109. (Numdam)
- [KM] S. Keel and S. Mori, Quotients by Groupoids, Ann. Math. 145 (1997), pp. 193-213. (JSTOR)
- [M] D. Mumford, Picard groups of moduli problems, Arithmetical Algebraic Geometry, 1963.
Homework
- Homework problems 1, due Apr 1, 2024.
- Homework problems 2, due Apr 24 (extended from Apr 17), 2024.
Topics for lectures given by participants
List of suggested topics (suitable for 2x45 minutes each).- (CMS) Coarse moduli spaces (Keel–Mori theorem, local structure of DM-stacks). [O, 11.1-3], [KM], [R]
- (ROOT) Root stacks and applications. Preferably after (CMS). [O, 10.3], [Cad, B17, B18]
- (TORIC) Toric stacks. Preferably after (ROOT) and (CMS). [BCS,FMN,GS]
- (GMS1) Good moduli spaces. Preferably after (CMS). [AlpGM], [Alp, 6.4], [HL, 14]
- (GMS2) Local structure of stacks. [AlpAA 3-4], [AHR], [Alp, 6.5-6], [HL, 13]
- (GMS3) Existence of good moduli spaces. [AHLH], [HL, 15]
- (AA) Artin's axioms for algebraicity. [AlpAA 1-2], [A4,A5], [HL, 12]
- (UNI) Uniformization of DM-stacks; topological and analytic stacks. [BN]
- (W) Weighted blow-ups. Useful to have (ROOT). [QR]
- (BRAUER) Gerbes, Brauer groups and quotient stacks. [O, 12], [EHKV, KV]
- Your favorite moduli stack / paper relating to moduli stacks.
Preliminary schedule
| # | Date | Place | Topic | Ref |
|---|---|---|---|---|
| 1 | Fri Jan 26, 10-12 | KTH 3721 | Functors and moduli problems | [O, 1.4-1.5], [BCEFFGK, 1] |
| 2 | Wed Jan 31, 15-17 | KTH 3721 | Grothendieck topologies, étale topology | [O, 1.3, 2.1-2.2] |
| 3 | Wed Feb 7, 15-17 | KTH 3418 | Fibered categories, groupoids | [O, 3] |
| 4 | Wed Feb 14, 15-17 | KTH 3418 | Descent and stacks | [O, 4] |
| 5 | Wed Feb 21, 15-17 | KTH 3418 | Stacks and torsors | [O, 4] |
| — | NO LECTURE | — | ||
| 6 | Wed Mar 6, 15-17 | KTH 3418 | Algebraic spaces: definitions and examples | [O, 5.1-5.3] |
| 7 | Wed Mar 13, 15-17 | KTH 3418 | Algebraic spaces: properties, points, quotients | [O, 5.4, 6] |
| 8 | *Mon Mar 18, 13-15 | KTH 3418 | Algebraic stacks | [O, 8] |
| 9 | Wed Mar 27, 15-17 | KTH 3418 | More on quotient stacks, Deligne–Mumford stacks | [O, 8] |
| — | NO LECTURE | — | Easter | |
| 10 | Wed Apr 10, 15-17 | KTH 3418 | Modular forms, Behrend's trace formula (Jan MF & Sjoerd) | [F1, F2, B, S] |
| 11 | Wed Apr 17, 15-17 | KTH 3418 | Root stacks (Jon-Magnus & Torger) | [O, 10.3], [Cad, QR22, B17, B18] |
| 12 | Wed Apr 24, 15-17 | KTH 3418 | Toric stacks (Darius & Ludvig) | [BCS, FMN, GS] |
| 13 | *Fri May 3, 10-12 | KTH 3418 | Good moduli spaces (Jarod Alper) | [Alp, 6.4-6] |
| 14 | Wed May 8, 15-17 | KTH 3418 | Topological stacks, uniformization (Benedetta & Josefien) | [BN] |
| — | NO LECTURE | — | Crafoord symposium | |
| 15 | *Mon May 20, 15-17 | KTH 3418 | Locally ringed topoi (Errol & Gabriel) | [SP, Tag 04KI], [SAG, 1.2], [DAG V, 1–2] |
| — | NO LECTURE | — | SMC master class | |
| 16 | *Tue Jun 4, 10-12 | KTH 3418 | Artin's axioms for algebraicity (Elmo & Giacomo) | [AlpAA 1-2], [A4, A5], [HL, 12] |