nLab Proper Morphisms, Completions, and the Grothendieck Existence Theorem

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• Jacob Lurie,

  Proper Morphisms, Completions, and the Grothendieck Existence Theorem

  (pdf)

on basic properties of E-∞ geometry

1. Generalities on spectral Deligne-Mumford stacks/algebraic spaces

1.1 Points of spectral Deligne-Mumford stacks

1.2 Étale morphisms

• étale morphism of schemes

1.3 Localic spectral Deligne-Mumford stacks

• algebraic space

1.4 Quasi-compactness of spectral Deligne-Mumford stacks

1.5 Local properties of spectral Deligne-Mumford stacks

2. Noetherian approximation

3. Properness

3.1 Strongly proper morphisms

3.2 The direct image theorem

3.3 Proper linear $\infty$-categories

• Wirthmüller context

3.4 Valuative criteria

4. Completions of modules

• torsion module, torsion approximation

• completion of a module

5. Completions of spectral Deligne-Mumford stacks

5.1 Formal completions

5.2 Truncations in $QCoh(\mathcal{X}_K^{\wedge})$

5.3 The Grothendieck existence theorem

• Grothendieck existence theorem

5.4 Algebraizability of formal stacks

6. Relationship with formal moduli problems

• formal moduli problem

A. Stone duality

• Stone duality

category: reference