86 Formal Algebraic Spaces

Section 86.1: Introduction
Section 86.2: Formal schemes à la EGA
- Equation 86.2.0.1
- Equation 86.2.0.2
- Lemma 86.2.1
- Lemma 86.2.2 slogan
- Remark 86.2.3: McQuillan's variant
- Remark 86.2.4: Sheafification of presheaves of topological spaces reference
Section 86.3: Conventions and notation
Section 86.4: Topological rings and modules
- Example 86.4.1
- Lemma 86.4.2
- Lemma 86.4.3
- Lemma 86.4.4
- Lemma 86.4.5 reference
- Lemma 86.4.6
- Definition 86.4.7
- Definition 86.4.8
- Lemma 86.4.9
- Lemma 86.4.10
- Lemma 86.4.11
- Lemma 86.4.12
Section 86.5: Taut ring maps
- Definition 86.5.1
- Lemma 86.5.2
- Lemma 86.5.3
- Lemma 86.5.4
- Lemma 86.5.5
- Lemma 86.5.6
- Lemma 86.5.7
- Lemma 86.5.8
- Lemma 86.5.9
- Lemma 86.5.10
Section 86.6: Adic ring maps
- Definition 86.6.1
- Lemma 86.6.2
- Lemma 86.6.3
- Lemma 86.6.4
- Lemma 86.6.5
Section 86.7: Weakly adic rings
- Definition 86.7.1 reference
- Lemma 86.7.2
- Lemma 86.7.3
- Lemma 86.7.4
- Lemma 86.7.5
- Lemma 86.7.6
Section 86.8: Descending properties
- Lemma 86.8.1
- Lemma 86.8.2
- Lemma 86.8.3
- Lemma 86.8.4
- Lemma 86.8.5
Section 86.9: Affine formal algebraic spaces
- Definition 86.9.1
- Lemma 86.9.2
- Lemma 86.9.3
- Lemma 86.9.4
- Lemma 86.9.5
- Lemma 86.9.6
- Definition 86.9.7
- Remark 86.9.8
- Definition 86.9.9
- Lemma 86.9.10
- Lemma 86.9.11
Section 86.10: Countably indexed affine formal algebraic spaces
- Lemma 86.10.1
- Definition 86.10.2
- Lemma 86.10.3
- Lemma 86.10.4
- Lemma 86.10.5
Section 86.11: Formal algebraic spaces
- Definition 86.11.1
- Lemma 86.11.2
- Lemma 86.11.3
- Remark 86.11.4
- Remark 86.11.5
Section 86.12: The reduction
- Lemma 86.12.1
- Example 86.12.2
- Lemma 86.12.3
- Lemma 86.12.4
Section 86.13: Colimits of algebraic spaces along thickenings
- Lemma 86.13.1
- Example 86.13.2
- Remark 86.13.3: Weak ideals of definition
- Remark 86.13.4: Ideals of definition
Section 86.14: Completion along a closed subset
- Lemma 86.14.1
- Lemma 86.14.2
- Definition 86.14.3
- Lemma 86.14.4
- Lemma 86.14.5
- Lemma 86.14.6
- Lemma 86.14.7 reference
Section 86.15: Fibre products
- Lemma 86.15.1
- Lemma 86.15.2
- Lemma 86.15.3
- Lemma 86.15.4
- Lemma 86.15.5
Section 86.16: Separation axioms for formal algebraic spaces
- Lemma 86.16.1
- Lemma 86.16.2
- Definition 86.16.3
- Lemma 86.16.4
- Lemma 86.16.5
Section 86.17: Quasi-compact formal algebraic spaces
- Lemma 86.17.1
- Definition 86.17.2
- Lemma 86.17.3
- Definition 86.17.4
- Lemma 86.17.5
Section 86.18: Quasi-compact and quasi-separated formal algebraic spaces
- Lemma 86.18.1 reference
- Remark 86.18.2
- Lemma 86.18.3
Section 86.19: Morphisms representable by algebraic spaces
- Lemma 86.19.1
- Lemma 86.19.2
- Lemma 86.19.3
- Lemma 86.19.4
- Lemma 86.19.5
- Lemma 86.19.6
- Lemma 86.19.7
- Lemma 86.19.8
- Lemma 86.19.9
- Lemma 86.19.10
- Example 86.19.11
- Remark 86.19.12: Warning
- Lemma 86.19.13
- Lemma 86.19.14
Section 86.20: Types of formal algebraic spaces
- Lemma 86.20.1
- Lemma 86.20.2
- Lemma 86.20.3
- Lemma 86.20.4
- Lemma 86.20.5
- Lemma 86.20.6
- Definition 86.20.7
- Lemma 86.20.8
- Remark 86.20.9: Warning
- Lemma 86.20.10
- Lemma 86.20.11
Section 86.21: Morphisms and continuous ring maps
- Lemma 86.21.1
- Situation 86.21.2
  - Equation 86.21.2.1
- Lemma 86.21.3
- Remark 86.21.4: Variant for adic-star
- Remark 86.21.5: Variant for Noetherian
- Situation 86.21.6
- Lemma 86.21.7
- Remark 86.21.8: Variant for adic-star
- Remark 86.21.9: Variant for Noetherian
- Remark 86.21.10: Another variant for Noetherian
- Situation 86.21.11
- Lemma 86.21.12
- Remark 86.21.13: Variant for adic-star
- Remark 86.21.14: Variant for Noetherian
- Situation 86.21.15
- Lemma 86.21.16
- Remark 86.21.17: Variant for adic-star
- Remark 86.21.18: Variant for Noetherian
Section 86.22: Taut ring maps and representability by algebraic spaces
- Lemma 86.22.1
- Lemma 86.22.2
Section 86.23: Adic morphisms
- Lemma 86.23.1
- Definition 86.23.2
Section 86.24: Morphisms of finite type
- Definition 86.24.1
- Lemma 86.24.2
- Lemma 86.24.3
- Lemma 86.24.4
- Lemma 86.24.5
- Lemma 86.24.6
- Example 86.24.7
- Lemma 86.24.8
- Lemma 86.24.9
Section 86.25: Surjective morphisms
- Definition 86.25.1
- Lemma 86.25.2
- Lemma 86.25.3
- Lemma 86.25.4
Section 86.26: Monomorphisms
- Definition 86.26.1
- Lemma 86.26.2
- Lemma 86.26.3
- Lemma 86.26.4
Section 86.27: Closed immersions
- Definition 86.27.1
- Lemma 86.27.2
- Lemma 86.27.3
- Lemma 86.27.4
- Lemma 86.27.5
- Example 86.27.6
- Lemma 86.27.7
Section 86.28: Restricted power series
- Remark 86.28.1: Universal property restricted power series reference
- Remark 86.28.2
Section 86.29: Algebras topologically of finite type
- Definition 86.29.1
- Lemma 86.29.2
- Lemma 86.29.3
- Remark 86.29.4
- Lemma 86.29.5
- Lemma 86.29.6
- Lemma 86.29.7
- Lemma 86.29.8
- Lemma 86.29.9
Section 86.30: Separation axioms for morphisms
- Definition 86.30.1
- Lemma 86.30.2
- Lemma 86.30.3
- Lemma 86.30.4
- Lemma 86.30.5
Section 86.31: Proper morphisms
- Definition 86.31.1
- Lemma 86.31.2
- Lemma 86.31.3
Section 86.32: Formal algebraic spaces and fpqc coverings
- Lemma 86.32.1 slogan
Section 86.33: Maps out of affine formal schemes
- Lemma 86.33.1
- Lemma 86.33.2
- Lemma 86.33.3
- Lemma 86.33.4
Section 86.34: The small étale site of a formal algebraic space
- Definition 86.34.1
  - Equation 86.34.1.1
- Lemma 86.34.2
- Lemma 86.34.3
- Lemma 86.34.4
- Lemma 86.34.5
- Lemma 86.34.6
- Lemma 86.34.7
Section 86.35: The structure sheaf
- Lemma 86.35.1
- Remark 86.35.2
- Lemma 86.35.3
- Remark 86.35.4
Section 86.36: Colimits of formal algebraic spaces
- Lemma 86.36.1
- Lemma 86.36.2
Section 86.37: Recompletion
- Lemma 86.37.1
- Lemma 86.37.2
- Definition 86.37.3
- Lemma 86.37.4
- Lemma 86.37.5
- Lemma 86.37.6
- Lemma 86.37.7
Section 86.38: Completion along a closed subspace
- Definition 86.38.1
- Lemma 86.38.2
- Lemma 86.38.3
- Lemma 86.38.4
- Lemma 86.38.5