Algebraic stacks
Spring 2017

---

This is a graduate level course in algebraic stacks on 7.5 credits.

Contents

• Fibered categories, descent and stacks
• Algebraic spaces: definition, points, sheaves, Chow's lemma, finiteness of cohomology
• Quotients of group actions
• Torsors
• Algebraic stacks
• Characterization of Deligne—Mumford stacks and algebraic spaces
• Gerbes, root stacks
• Coarse moduli spaces
• Artin approximation and Artin's axioms for algebraicity
• Local structure of Artin stacks

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]). Grothendieck topologies (see [O, Ch. 2.1—2.2] and [V1]).

Examiner

David Rydh

Course start

First meeting on Friday, Feb 24, 10:15-12:00, KTH 3418.

Time and place

After that, mostly Wednesdays, 15:15–17:00 (preliminary). Alternating between KTH 3721 or SU ???, depending on where the algebra seminar is.

Examination

Lectures will mainly be given by participants. All participants should study the material, including exercises, carefully before each meeting. The lecturer should summarize main definitions, theorems, proofs, examples and exercises but we will also try to have some more informal discussions in class.

Literature

Main text-book:

• [O] M. Olsson, Algebraic stacks, 2016 (AMS Bookstore)

Other references

• [Alp] 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. (version on Kresch web page)
• [K] D. Knutson, Algebraic spaces, 1971. (Springer Link)
• [LMB] G. Laumon & L. Moret-Bailly, Champs Algébriques, 2000.
• [V1] A. Vistoli, Grothendieck topologies, fibered categories and descent theory, FGA Explained, Part 1, (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)

Articles

• [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)
• [AHR] J. Alper, J. Hall & D. Rydh, A Luna étale slice theorem for algebraic stacks, 2015. (arXiv:1504.06467)
• [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)
• [HR] J. Hall & D. Rydh, Artin's criteria for algebraicity revisited, 2015. (preprint)
• [M] D. Mumford, Picard groups of moduli problems, Arithmetical Algebraic Geometry, 1963.
• [R] D. Rydh, Existence and properties of geometric quotients, J. Algebraic Geom. (2013), p. 629-669. (arXiv:0708.3333)

Preliminary schedule

#	Date	Place	Lecturer	Topic	Ref
1	Feb 24, 10-12	KTH 3418	David Rydh	Introduction (examples, groupoids, atlas, stacks)	[BCEFFGK, 1]
2	Mar 1, 15-17	KTH 3721	Magnus Carlson	Descent	[O, 4.1—4, 4A, 4C], [BCEFFGK, App. A]
3	Mar 8, 15-17	SU 5:22	Eric Ahlqvist	Torsors, stacks, fiber products, examples	[O, 4.5—6, 3.4.9], [BCEFFGK, 2.5, 4]
4	Mar 15, 15:40-17	KTH 3418	Magnus Carlson	Algebraic spaces. Examples.	[O, 5], [K, 0, 2.1-2.3]
5	Mar 22, 14:30-16:30	SU 5:22	David Rydh	Quotients by finite groups on schemes. Points of algebraic spaces.	[O, 6], [R, 4]
-	Mar 29	---	---	No lecture
6	Apr 5, 10-12	KTH 3721	Magnus Carlson	Algebraic stacks. Examples (quotient stacks, classifying stacks, quotients of groupoids).	[O, 8.1-2,8.4, 8.E]
7	Apr 12, 15-17	KTH 3418	Eric Ahlqvist	Deligne—Mumford stacks, inertia stacks. Examples.	[O, 8.3-4, 8.B], [LMB, 8]
8	Apr 19, 14:30-16:30	SU 5:32	Eric/David	Inertia of smooth DM-stacks. Quasi-coherent sheaves. Generators.	[O, 8.B, 7.1, 9.1], [LMB, 12, 13, 15.4]
9	Apr 21, 13:15-15:00	KTH 3721	Magnus Carlson	Coarse moduli spaces.	[O, 11], [R, 6]
10	Apr 26	KTH 3418	Eric Ahlqvist	Root stacks. Gerbes.	[O, 10.3, 12], [V2]
11	May 3	SU 5:34	David Rydh	Flat presentations, rigidification, gerbes, residual gerbes, root gerbes. (Zariski's main theorem, Chow Lemmas, valuative criteria.)	[A4, 6], [V2], [O, 7.4, 11.4], [LMB, 10.1, 16.6]
12	May 10	KTH 3418	Magnus Carlson	Artin approximation	[Alp, 1.1-1.4], [A3]
13	May 17, 13:15-15:00	SU 5:22	Eric Ahlqvist	Artin algebraization. Artin's axioms.	[Alp, 2], [A4, A5]
14	May 24, 15:15-17:00	KTH 3418	Magnus Carlson	Quotient stacks.	[Alp, 3]
-	May 31	---	---	No lecture
-	Jun 7	---	---	No lecture
-	Jun 14	---	---	No lecture
15	Jun 19, 13:15-15:00	KTH 3418	David Rydh	Quotient stacks.	[Alp, 4], [AHR]
16	Jun 19, 15:15-17:00	KTH 3418	Eric Ahlqvist	Artin's axioms.	[A4, A5, HR]

Department of Mathematics — Royal Institute of Technology (KTH)