75 Groupoids in Algebraic Spaces

Section 75.1: Introduction
Section 75.2: Conventions
Section 75.3: Notation
Section 75.4: Equivalence relations
- Definition 75.4.1
- Lemma 75.4.2
- Definition 75.4.3
- Lemma 75.4.4
Section 75.5: Group algebraic spaces
- Definition 75.5.1
- Lemma 75.5.2
Section 75.6: Properties of group algebraic spaces
- Lemma 75.6.1
- Lemma 75.6.2
- Lemma 75.6.3
Section 75.7: Examples of group algebraic spaces
- Example 75.7.1: Multiplicative group algebraic space
- Example 75.7.2: Roots of unity as a group algebraic space
- Example 75.7.3: Additive group algebraic space
- Example 75.7.4: General linear group algebraic space
- Example 75.7.5
- Example 75.7.6: Constant group algebraic space
Section 75.8: Actions of group algebraic spaces
- Definition 75.8.1
  - Equation 75.8.1.1
- Definition 75.8.2
- Lemma 75.8.3
Section 75.9: Principal homogeneous spaces
- Definition 75.9.1
- Lemma 75.9.2
- Definition 75.9.3
- Lemma 75.9.4
- Lemma 75.9.5
Section 75.10: Equivariant quasi-coherent sheaves
- Definition 75.10.1
- Lemma 75.10.2
Section 75.11: Groupoids in algebraic spaces
- Definition 75.11.1
- Lemma 75.11.2
- Lemma 75.11.3
- Lemma 75.11.4
- Lemma 75.11.5
  - Equation 75.11.5.1
- Lemma 75.11.6
Section 75.12: Quasi-coherent sheaves on groupoids
- Definition 75.12.1
- Lemma 75.12.2
- Lemma 75.12.3
- Lemma 75.12.4
- Lemma 75.12.5
Section 75.13: Crystals in quasi-coherent sheaves
- Lemma 75.13.1
- Lemma 75.13.2
Section 75.14: Groupoids and group spaces
- Lemma 75.14.1
- Lemma 75.14.2
Section 75.15: The stabilizer group algebraic space
- Lemma 75.15.1
- Definition 75.15.2
- Lemma 75.15.3
Section 75.16: Restricting groupoids
- Lemma 75.16.1
- Definition 75.16.2
- Lemma 75.16.3
Section 75.17: Invariant subspaces
- Definition 75.17.1
- Lemma 75.17.2
Section 75.18: Quotient sheaves
- Equation 75.18.0.1
- Definition 75.18.1
- Remark 75.18.2
- Definition 75.18.3
- Lemma 75.18.4
- Lemma 75.18.5
- Lemma 75.18.6
- Lemma 75.18.7
Section 75.19: Quotient stacks
- Equation 75.19.0.1
- Definition 75.19.1
- Lemma 75.19.2
- Lemma 75.19.3
- Remark 75.19.4
Section 75.20: Functoriality of quotient stacks
- Lemma 75.20.1
- Lemma 75.20.2
Section 75.21: The 2-cartesian square of a quotient stack
- Lemma 75.21.1
- Lemma 75.21.2
- Lemma 75.21.3
Section 75.22: The 2-coequalizer property of a quotient stack
- Lemma 75.22.1
- Lemma 75.22.2
Section 75.23: Explicit description of quotient stacks
- Lemma 75.23.1
Section 75.24: Restriction and quotient stacks
- Lemma 75.24.1
- Lemma 75.24.2
- Lemma 75.24.3
Section 75.25: Inertia and quotient stacks
- Lemma 75.25.1
- Lemma 75.25.2
Section 75.26: Gerbes and quotient stacks
- Lemma 75.26.1
- Lemma 75.26.2
Section 75.27: Quotient stacks and change of big site
- Lemma 75.27.1
Section 75.28: Separation conditions
- Lemma 75.28.1
- Lemma 75.28.2