87 Formal Algebraic Spaces

Section 87.1: Introduction
Section 87.2: Formal schemes à la EGA
- Equation 87.2.0.1
- Equation 87.2.0.2
- Lemma 87.2.1
- Lemma 87.2.2 slogan
- Remark 87.2.3: McQuillan's variant
- Remark 87.2.4: Sheafification of presheaves of topological spaces reference
Section 87.3: Conventions and notation
Section 87.4: Topological rings and modules
- Example 87.4.1
- Lemma 87.4.2
- Lemma 87.4.3
- Lemma 87.4.4
- Lemma 87.4.5 reference
- Lemma 87.4.6
- Definition 87.4.7
- Definition 87.4.8
- Lemma 87.4.9
- Lemma 87.4.10
- Lemma 87.4.11
- Lemma 87.4.12
Section 87.5: Taut ring maps
- Definition 87.5.1
- Lemma 87.5.2
- Lemma 87.5.3
- Lemma 87.5.4
- Lemma 87.5.5
- Lemma 87.5.6
- Lemma 87.5.7
- Lemma 87.5.8
- Lemma 87.5.9
- Lemma 87.5.10
Section 87.6: Adic ring maps
- Definition 87.6.1
- Lemma 87.6.2
- Lemma 87.6.3
- Lemma 87.6.4
- Lemma 87.6.5
Section 87.7: Weakly adic rings
- Definition 87.7.1 reference
- Lemma 87.7.2
- Lemma 87.7.3
- Lemma 87.7.4
- Lemma 87.7.5
- Lemma 87.7.6
Section 87.8: Descending properties
- Lemma 87.8.1
- Lemma 87.8.2
- Lemma 87.8.3
- Lemma 87.8.4
- Lemma 87.8.5
Section 87.9: Affine formal algebraic spaces
- Definition 87.9.1
- Lemma 87.9.2
- Lemma 87.9.3
- Lemma 87.9.4
- Lemma 87.9.5
- Lemma 87.9.6
- Definition 87.9.7
- Remark 87.9.8
- Definition 87.9.9
- Lemma 87.9.10
- Lemma 87.9.11
Section 87.10: Countably indexed affine formal algebraic spaces
- Lemma 87.10.1
- Definition 87.10.2
- Lemma 87.10.3
- Lemma 87.10.4
- Lemma 87.10.5
Section 87.11: Formal algebraic spaces
- Definition 87.11.1
- Lemma 87.11.2
- Lemma 87.11.3
- Remark 87.11.4
- Remark 87.11.5
Section 87.12: The reduction
- Lemma 87.12.1
- Example 87.12.2
- Lemma 87.12.3
- Lemma 87.12.4
Section 87.13: Colimits of algebraic spaces along thickenings
- Lemma 87.13.1
- Example 87.13.2
- Remark 87.13.3: Weak ideals of definition
- Remark 87.13.4: Ideals of definition
Section 87.14: Completion along a closed subset
- Lemma 87.14.1
- Lemma 87.14.2
- Definition 87.14.3
- Lemma 87.14.4
- Lemma 87.14.5
- Lemma 87.14.6
- Lemma 87.14.7 reference
Section 87.15: Fibre products
- Lemma 87.15.1
- Lemma 87.15.2
- Lemma 87.15.3
- Lemma 87.15.4
- Lemma 87.15.5
Section 87.16: Separation axioms for formal algebraic spaces
- Lemma 87.16.1
- Lemma 87.16.2
- Definition 87.16.3
- Lemma 87.16.4
- Lemma 87.16.5
Section 87.17: Quasi-compact formal algebraic spaces
- Lemma 87.17.1
- Definition 87.17.2
- Lemma 87.17.3
- Definition 87.17.4
- Lemma 87.17.5
Section 87.18: Quasi-compact and quasi-separated formal algebraic spaces
- Lemma 87.18.1 reference
- Remark 87.18.2
- Lemma 87.18.3
Section 87.19: Morphisms representable by algebraic spaces
- Lemma 87.19.1
- Lemma 87.19.2
- Lemma 87.19.3
- Lemma 87.19.4
- Lemma 87.19.5
- Lemma 87.19.6
- Lemma 87.19.7
- Lemma 87.19.8
- Lemma 87.19.9
- Lemma 87.19.10
- Example 87.19.11
- Remark 87.19.12: Warning
- Lemma 87.19.13
- Lemma 87.19.14
Section 87.20: Types of formal algebraic spaces
- Lemma 87.20.1
- Lemma 87.20.2
- Lemma 87.20.3
- Lemma 87.20.4
- Lemma 87.20.5
- Lemma 87.20.6
- Definition 87.20.7
- Lemma 87.20.8
- Remark 87.20.9: Warning
- Lemma 87.20.10
- Lemma 87.20.11
Section 87.21: Morphisms and continuous ring maps
- Lemma 87.21.1
- Situation 87.21.2
  - Equation 87.21.2.1
- Lemma 87.21.3
- Remark 87.21.4: Variant for adic-star
- Remark 87.21.5: Variant for Noetherian
- Situation 87.21.6
- Lemma 87.21.7
- Remark 87.21.8: Variant for adic-star
- Remark 87.21.9: Variant for Noetherian
- Remark 87.21.10: Another variant for Noetherian
- Situation 87.21.11
- Lemma 87.21.12
- Remark 87.21.13: Variant for adic-star
- Remark 87.21.14: Variant for Noetherian
- Situation 87.21.15
- Lemma 87.21.16
- Remark 87.21.17: Variant for adic-star
- Remark 87.21.18: Variant for Noetherian
Section 87.22: Taut ring maps and representability by algebraic spaces
- Lemma 87.22.1
- Lemma 87.22.2
Section 87.23: Adic morphisms
- Lemma 87.23.1
- Definition 87.23.2
Section 87.24: Morphisms of finite type
- Definition 87.24.1
- Lemma 87.24.2
- Lemma 87.24.3
- Lemma 87.24.4
- Lemma 87.24.5
- Lemma 87.24.6
- Example 87.24.7
- Lemma 87.24.8
- Lemma 87.24.9
Section 87.25: Surjective morphisms
- Definition 87.25.1
- Lemma 87.25.2
- Lemma 87.25.3
- Lemma 87.25.4
Section 87.26: Monomorphisms
- Definition 87.26.1
- Lemma 87.26.2
- Lemma 87.26.3
- Lemma 87.26.4
Section 87.27: Closed immersions
- Definition 87.27.1
- Lemma 87.27.2
- Lemma 87.27.3
- Lemma 87.27.4
- Lemma 87.27.5
- Example 87.27.6
- Lemma 87.27.7
Section 87.28: Restricted power series
- Remark 87.28.1: Universal property restricted power series reference
- Remark 87.28.2
Section 87.29: Algebras topologically of finite type
- Definition 87.29.1
- Lemma 87.29.2
- Lemma 87.29.3
- Remark 87.29.4
- Lemma 87.29.5
- Lemma 87.29.6
- Lemma 87.29.7
- Lemma 87.29.8
- Lemma 87.29.9
Section 87.30: Separation axioms for morphisms
- Definition 87.30.1
- Lemma 87.30.2
- Lemma 87.30.3
- Lemma 87.30.4
- Lemma 87.30.5
Section 87.31: Proper morphisms
- Definition 87.31.1
- Lemma 87.31.2
- Lemma 87.31.3
Section 87.32: Formal algebraic spaces and fpqc coverings
- Lemma 87.32.1 slogan
Section 87.33: Maps out of affine formal schemes
- Lemma 87.33.1
- Lemma 87.33.2
- Lemma 87.33.3
- Lemma 87.33.4
Section 87.34: The small étale site of a formal algebraic space
- Definition 87.34.1
  - Equation 87.34.1.1
- Lemma 87.34.2
- Lemma 87.34.3
- Lemma 87.34.4
- Lemma 87.34.5
- Lemma 87.34.6
- Lemma 87.34.7
Section 87.35: The structure sheaf
- Lemma 87.35.1
- Remark 87.35.2
- Lemma 87.35.3
- Remark 87.35.4
Section 87.36: Colimits of formal algebraic spaces
- Lemma 87.36.1
- Lemma 87.36.2
Section 87.37: Recompletion
- Lemma 87.37.1
- Lemma 87.37.2
- Definition 87.37.3
- Lemma 87.37.4
- Lemma 87.37.5
- Lemma 87.37.6
- Lemma 87.37.7
Section 87.38: Completion along a closed subspace
- Definition 87.38.1
- Lemma 87.38.2
- Lemma 87.38.3
- Lemma 87.38.4
- Lemma 87.38.5