This is a graduate level course on algebraic stacks (7.5 credits). Algebraic stacks is the natural language of moduli theory. Whenever parametrizing algebraic objects with automorphisms, stacks are necessary for a full and well-behaved description. From another perspective, stacks arise when we consider quotients of groups acting on schemes. This course will give an introduction to stacks and highlight some topics.

Preliminary contents


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.


Stack 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).


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).


As the main text-book we will use Martin Olsson's book. It is more focusing on foundations than examples though and I will occasionally refer to other sources. Some other useful books and notes: Articles:


Topics for lectures given by participants

List of suggested topics (suitable for 2x45 minutes each).

Preliminary schedule

# Date Place Topic Ref
1Fri Jan 26, 10-12KTH 3721Functors and moduli problems[O, 1.4-1.5],
2Wed Jan 31, 15-17KTH 3721Grothendieck topologies, étale topology[O, 1.3, 2.1-2.2]
3Wed Feb 7, 15-17KTH 3418Fibered categories, groupoids[O, 3]
4Wed Feb 14, 15-17KTH 3418Descent and stacks[O, 4]
5Wed Feb 21, 15-17KTH 3418Stacks and torsors[O, 4]
6Wed Mar 6, 15-17KTH 3418Algebraic spaces: definitions and examples[O, 5.1-5.3]
7Wed Mar 13, 15-17KTH 3418Algebraic spaces: properties, points, quotients[O, 5.4, 6]
8*Mon Mar 18, 13-15KTH 3418Algebraic stacks[O, 8]
9Wed Mar 27, 15-17KTH 3418More on quotient stacks, Deligne–Mumford stacks[O, 8]
10Wed Apr 10, 15-17KTH 3418Modular forms, Behrend's trace formula
(Jan MF & Sjoerd)
[F1, F2, B, S]
11Wed Apr 17, 15-17KTH 3418Root stacks (Jon-Magnus & Torger)[O, 10.3],
[Cad, QR22, B17, B18]
12Wed Apr 24, 15-17KTH 3418Toric stacks (Darius & Ludvig)[BCS, FMN, GS]
13*Fri May 3, 10-12KTH 3418Good moduli spaces (Jarod Alper)[Alp, 6.4-6]
14Wed May 8, 15-17KTH 3418Topological stacks, uniformization
(Benedetta & Josefien)
NO LECTURECrafoord symposium
15*Mon May 20, 15-17KTH 3418Locally ringed topoi (Errol & Gabriel)[SP, Tag 04KI],
[SAG, 1.2],
[DAG V, 1–2]
NO LECTURESMC master class
16*Tue Jun 4, 10-12KTH 3418Artin's axioms for algebraicity (Elmo & Giacomo)[AlpAA 1-2],
[A4, A5], [HL, 12]

Further topics that we will probably not discuss: