78 Groupoids in Algebraic Spaces

• Section 78.1: Introduction
• Section 78.2: Conventions
• Section 78.3: Notation
• Section 78.4: Equivalence relations
  • Definition 78.4.1
  • Lemma 78.4.2
  • Definition 78.4.3
  • Lemma 78.4.4
• Section 78.5: Group algebraic spaces
  • Definition 78.5.1
  • Lemma 78.5.2
• Section 78.6: Properties of group algebraic spaces
  • Lemma 78.6.1
  • Lemma 78.6.2
  • Lemma 78.6.3
• Section 78.7: Examples of group algebraic spaces
  • Example 78.7.1: Multiplicative group algebraic space
  • Example 78.7.2: Roots of unity as a group algebraic space
  • Example 78.7.3: Additive group algebraic space
  • Example 78.7.4: General linear group algebraic space
  • Example 78.7.5
  • Example 78.7.6: Constant group algebraic space
• Section 78.8: Actions of group algebraic spaces
  • Definition 78.8.1
    • Equation 78.8.1.1
  • Definition 78.8.2
  • Lemma 78.8.3
• Section 78.9: Principal homogeneous spaces
  • Definition 78.9.1
  • Lemma 78.9.2
  • Definition 78.9.3
  • Lemma 78.9.4
  • Lemma 78.9.5
• Section 78.10: Equivariant quasi-coherent sheaves
  • Definition 78.10.1
  • Lemma 78.10.2
• Section 78.11: Groupoids in algebraic spaces
  • Definition 78.11.1
  • Lemma 78.11.2
  • Lemma 78.11.3
  • Lemma 78.11.4
  • Lemma 78.11.5
    • Equation 78.11.5.1
  • Lemma 78.11.6
• Section 78.12: Quasi-coherent sheaves on groupoids
  • Definition 78.12.1
  • Lemma 78.12.2
  • Lemma 78.12.3
  • Lemma 78.12.4
  • Lemma 78.12.5
  • Lemma 78.12.6
• Section 78.13: Colimits of quasi-coherent modules
  • Lemma 78.13.1
  • Remark 78.13.2
  • Lemma 78.13.3
  • Lemma 78.13.4
• Section 78.14: Crystals in quasi-coherent sheaves
  • Lemma 78.14.1
  • Lemma 78.14.2
• Section 78.15: Groupoids and group spaces
  • Lemma 78.15.1
  • Lemma 78.15.2
• Section 78.16: The stabilizer group algebraic space
  • Lemma 78.16.1
  • Definition 78.16.2
  • Lemma 78.16.3
• Section 78.17: Restricting groupoids
  • Lemma 78.17.1
  • Definition 78.17.2
  • Lemma 78.17.3
• Section 78.18: Invariant subspaces
  • Definition 78.18.1
  • Lemma 78.18.2
• Section 78.19: Quotient sheaves
  • Equation 78.19.0.1
  • Definition 78.19.1
  • Remark 78.19.2
  • Definition 78.19.3
  • Lemma 78.19.4
  • Lemma 78.19.5
  • Lemma 78.19.6
  • Lemma 78.19.7
• Section 78.20: Quotient stacks
  • Equation 78.20.0.1
  • Definition 78.20.1
  • Lemma 78.20.2
  • Lemma 78.20.3
  • Remark 78.20.4
• Section 78.21: Functoriality of quotient stacks
  • Lemma 78.21.1
  • Lemma 78.21.2
• Section 78.22: The 2-cartesian square of a quotient stack
  • Lemma 78.22.1
  • Lemma 78.22.2
  • Lemma 78.22.3
• Section 78.23: The 2-coequalizer property of a quotient stack
  • Lemma 78.23.1
  • Lemma 78.23.2
• Section 78.24: Explicit description of quotient stacks
  • Lemma 78.24.1
• Section 78.25: Restriction and quotient stacks
  • Lemma 78.25.1
  • Lemma 78.25.2
  • Lemma 78.25.3
• Section 78.26: Inertia and quotient stacks
  • Lemma 78.26.1
  • Lemma 78.26.2
• Section 78.27: Gerbes and quotient stacks
  • Lemma 78.27.1
  • Lemma 78.27.2
• Section 78.28: Quotient stacks and change of big site
  • Lemma 78.28.1
• Section 78.29: Separation conditions
  • Lemma 78.29.1
  • Lemma 78.29.2