77 Groupoids in Algebraic Spaces

Section 77.1: Introduction
Section 77.2: Conventions
Section 77.3: Notation
Section 77.4: Equivalence relations
- Definition 77.4.1
- Lemma 77.4.2
- Definition 77.4.3
- Lemma 77.4.4
Section 77.5: Group algebraic spaces
- Definition 77.5.1
- Lemma 77.5.2
Section 77.6: Properties of group algebraic spaces
- Lemma 77.6.1
- Lemma 77.6.2
- Lemma 77.6.3
Section 77.7: Examples of group algebraic spaces
- Example 77.7.1: Multiplicative group algebraic space
- Example 77.7.2: Roots of unity as a group algebraic space
- Example 77.7.3: Additive group algebraic space
- Example 77.7.4: General linear group algebraic space
- Example 77.7.5
- Example 77.7.6: Constant group algebraic space
Section 77.8: Actions of group algebraic spaces
- Definition 77.8.1
  - Equation 77.8.1.1
- Definition 77.8.2
- Lemma 77.8.3
Section 77.9: Principal homogeneous spaces
- Definition 77.9.1
- Lemma 77.9.2
- Definition 77.9.3
- Lemma 77.9.4
- Lemma 77.9.5
Section 77.10: Equivariant quasi-coherent sheaves
- Definition 77.10.1
- Lemma 77.10.2
Section 77.11: Groupoids in algebraic spaces
- Definition 77.11.1
- Lemma 77.11.2
- Lemma 77.11.3
- Lemma 77.11.4
- Lemma 77.11.5
  - Equation 77.11.5.1
- Lemma 77.11.6
Section 77.12: Quasi-coherent sheaves on groupoids
- Definition 77.12.1
- Lemma 77.12.2
- Lemma 77.12.3
- Lemma 77.12.4
- Lemma 77.12.5
- Lemma 77.12.6
Section 77.13: Colimits of quasi-coherent modules
- Lemma 77.13.1
- Remark 77.13.2
- Lemma 77.13.3
- Lemma 77.13.4
Section 77.14: Crystals in quasi-coherent sheaves
- Lemma 77.14.1
- Lemma 77.14.2
Section 77.15: Groupoids and group spaces
- Lemma 77.15.1
- Lemma 77.15.2
Section 77.16: The stabilizer group algebraic space
- Lemma 77.16.1
- Definition 77.16.2
- Lemma 77.16.3
Section 77.17: Restricting groupoids
- Lemma 77.17.1
- Definition 77.17.2
- Lemma 77.17.3
Section 77.18: Invariant subspaces
- Definition 77.18.1
- Lemma 77.18.2
Section 77.19: Quotient sheaves
- Equation 77.19.0.1
- Definition 77.19.1
- Remark 77.19.2
- Definition 77.19.3
- Lemma 77.19.4
- Lemma 77.19.5
- Lemma 77.19.6
- Lemma 77.19.7
Section 77.20: Quotient stacks
- Equation 77.20.0.1
- Definition 77.20.1
- Lemma 77.20.2
- Lemma 77.20.3
- Remark 77.20.4
Section 77.21: Functoriality of quotient stacks
- Lemma 77.21.1
- Lemma 77.21.2
Section 77.22: The 2-cartesian square of a quotient stack
- Lemma 77.22.1
- Lemma 77.22.2
- Lemma 77.22.3
Section 77.23: The 2-coequalizer property of a quotient stack
- Lemma 77.23.1
- Lemma 77.23.2
Section 77.24: Explicit description of quotient stacks
- Lemma 77.24.1
Section 77.25: Restriction and quotient stacks
- Lemma 77.25.1
- Lemma 77.25.2
- Lemma 77.25.3
Section 77.26: Inertia and quotient stacks
- Lemma 77.26.1
- Lemma 77.26.2
Section 77.27: Gerbes and quotient stacks
- Lemma 77.27.1
- Lemma 77.27.2
Section 77.28: Quotient stacks and change of big site
- Lemma 77.28.1
Section 77.29: Separation conditions
- Lemma 77.29.1
- Lemma 77.29.2