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



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]).


Stack 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.


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.


Main text-book: Other references Articles

Preliminary schedule

# Date Place Lecturer Topic Ref
1Feb 24, 10-12KTH 3418David RydhIntroduction (examples, groupoids, atlas, stacks)[BCEFFGK, 1]
2Mar 1, 15-17KTH 3721Magnus CarlsonDescent[O, 4.1—4, 4A, 4C],
3Mar 8, 15-17SU 5:22Eric AhlqvistTorsors, stacks, fiber products, examples[O, 4.5—6, 3.4.9],
[BCEFFGK, 2.5, 4]
4Mar 15, 15:40-17KTH 3418Magnus CarlsonAlgebraic spaces. Examples.[O, 5], [K, 0, 2.1-2.3]
5Mar 22, 14:30-16:30SU 5:22David RydhQuotients by finite groups on schemes. Points of algebraic spaces.[O, 6], [R, 4]
-Mar 29------No lecture
6Apr 5, 10-12KTH 3721Magnus CarlsonAlgebraic stacks. Examples (quotient stacks, classifying stacks, quotients of groupoids).[O, 8.1-2,8.4, 8.E]
7Apr 12, 15-17KTH 3418Eric AhlqvistDeligne—Mumford stacks, inertia stacks. Examples.[O, 8.3-4, 8.B], [LMB, 8]
8Apr 19, 14:30-16:30SU 5:32Eric/DavidInertia of smooth DM-stacks. Quasi-coherent sheaves. Generators.[O, 8.B, 7.1, 9.1], [LMB, 12, 13, 15.4]
9Apr 21, 13:15-15:00KTH 3721Magnus CarlsonCoarse moduli spaces.[O, 11], [R, 6]
10Apr 26KTH 3418Eric AhlqvistRoot stacks. Gerbes.[O, 10.3, 12], [V2]
11May 3SU 5:34David RydhFlat 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]
12May 10KTH 3418Magnus CarlsonArtin approximation[Alp, 1.1-1.4], [A3]
13May 17, 13:15-15:00SU 5:22Eric AhlqvistArtin algebraization. Artin's axioms.[Alp, 2], [A4, A5]
14May 24, 15:15-17:00KTH 3418Magnus CarlsonQuotient stacks.[Alp, 3]
-May 31------No lecture
-Jun 7------No lecture
-Jun 14------No lecture
15Jun 19, 13:15-15:00KTH 3418David RydhQuotient stacks.[Alp, 4], [AHR]
16Jun 19, 15:15-17:00KTH 3418Eric AhlqvistArtin's axioms.[A4, A5, HR]

We will probably only briefly introduce / not cover the following topics: