Local structure of algebraic stacks

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Supporting material to the lecture series on the local structure of algebraic stacks for School on Motives and Stacks in Essen, Sep 23–27, 2019.

Outline of lectures

• Lecture 1: Quotients stacks and good moduli spaces
  Refs: [Alp13, §1–4] ([KM97], [Ryd13])
• Lecture 2: The local structure of algebraic stacks
  Refs: [AHR1, §1], [AHR2, §1, §3] ([Ols06])
• Lecture 3: Deformation theory and Coherent completeness
  Refs: [AHR1, §2], [AHR2, §4–5] ([HR19])
• Lecture 4: Tannaka duality, Artin approximation and equivariant Artin algebraization
  Refs: [AHR1, §A, §3], [AHR2, §12] ([Art69a], [Art69b])
• Lecture 5: Applications

Exercises

• Exercises 1
• Exercises 2
• Exercises 3

Lecture notes

• Lecture 1: we covered 1.1–1.8 (only definition), we'll continue with 1.8.
• Lecture 2: we covered 2.1–2.6, we'll continue with 2.7.
• Lecture 3
• Lecture 4
• Lecture 5

Main References

• [Alp13] J. Alper, Good moduli spaces for Artin stacks, Ann. Inst. Fourier 63 (2013), no. 6, 2349–2402 (Numdam Open Access)
• [AHR1] J. Alper, J. Hall & D. Rydh, A Luna étale slice theorem for algebraic stacks, 2015. (version as of Sep 20, 2019)
• [AHR2] J. Alper, J. Hall & D. Rydh, The étale local structure of algebraic stacks, Preprint Sep 2019. (version as of Sep 18, 2019)
• [AHHR3] J. Alper, J. Hall, D. Halpern-Leistner & D. Rydh, Artin algebraization for pairs and applications to the local structure of stacks and Ferrand pushouts, in preparation.

Other references

• [Alp14] J. Alper, Adequate moduli spaces and geometrically reductive group schemes, Algebr. Geom. 1 (2014), no. 4, 489–531 (Open Access)
• [Alp15] J. Alper, Artin algebraization and quotient stacks, Lecture notes from summer school in Mainz, 2015. (arXiv:1510.07804)
• [Art69a] M. Artin, Algebraic approximation of structures over complete local rings, Publications mathématiques de l'IHÉS, 36 (1969), 23–58. (Numdam Open Access)
• [Art69b] M. Artin, Algebraization of formal moduli I, 1969
• [HR19] J. Hall & D. Rydh, Coherent Tannaka duality and algebraicity of Hom-stacks, Algebra Number Theory (2019), to appear. (latest version, to be updated)
• [KM97] S. Keel & S. Mori, Quotients by groupoids, Ann. of Math. (2) 145 (1997), no. 1, 193–213. (JSTOR)
• [Ols06] M. Olsson, Deformation theory of representable morphisms of algebraic stacks, Math. Z. 253 (2006), no. 1, 25–62. (SpringerLink)
• [Ryd13] D. Rydh, Existence and properties of geometric quotients, J. Algebraic Geom. (2013), 629–669. (arXiv:0708.3333)

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