81 Formal Algebraic Spaces

Section 81.1: Introduction
Section 81.2: Formal schemes à la EGA
- Equation 81.2.0.1
- Equation 81.2.0.2
- Lemma 81.2.1
- Lemma 81.2.2 slogan
- Remark 81.2.3: McQuillan's variant
- Remark 81.2.4: Sheafification of presheaves of topological spaces reference
Section 81.3: Conventions and notation
Section 81.4: Topological rings and modules
- Example 81.4.1
- Lemma 81.4.2
- Lemma 81.4.3
- Lemma 81.4.4
- Lemma 81.4.5 reference
- Lemma 81.4.6
- Definition 81.4.7
- Definition 81.4.8
- Lemma 81.4.9
- Lemma 81.4.10
- Definition 81.4.11
- Lemma 81.4.12
- Lemma 81.4.13
- Lemma 81.4.14
- Lemma 81.4.15
Section 81.5: Affine formal algebraic spaces
- Definition 81.5.1
- Lemma 81.5.2
- Lemma 81.5.3
- Lemma 81.5.4
- Lemma 81.5.5
- Lemma 81.5.6
- Definition 81.5.7
- Remark 81.5.8
- Definition 81.5.9
- Lemma 81.5.10
- Lemma 81.5.11
Section 81.6: Countably indexed affine formal algebraic spaces
- Lemma 81.6.1
- Definition 81.6.2
- Lemma 81.6.3
- Lemma 81.6.4
- Lemma 81.6.5
Section 81.7: Formal algebraic spaces
- Definition 81.7.1
- Lemma 81.7.2
- Lemma 81.7.3
- Lemma 81.7.4
- Remark 81.7.5
- Remark 81.7.6
Section 81.8: Colimits of algebraic spaces along thickenings
- Lemma 81.8.1
- Example 81.8.2
- Remark 81.8.3: Weak ideals of definition
- Remark 81.8.4: Ideals of definition
Section 81.9: Completion along a closed subset
- Lemma 81.9.1
- Lemma 81.9.2
- Definition 81.9.3
- Lemma 81.9.4
- Lemma 81.9.5 reference
Section 81.10: Fibre products
- Lemma 81.10.1
- Lemma 81.10.2
- Lemma 81.10.3
- Lemma 81.10.4
- Lemma 81.10.5
Section 81.11: Separation axioms for formal algebraic spaces
- Lemma 81.11.1
- Lemma 81.11.2
- Definition 81.11.3
- Lemma 81.11.4
- Lemma 81.11.5
Section 81.12: Quasi-compact formal algebraic spaces
- Lemma 81.12.1
- Definition 81.12.2
- Lemma 81.12.3
- Definition 81.12.4
- Lemma 81.12.5
Section 81.13: Quasi-compact and quasi-separated formal algebraic spaces
- Lemma 81.13.1 reference
- Remark 81.13.2
- Lemma 81.13.3
Section 81.14: Morphisms representable by algebraic spaces
- Lemma 81.14.1
- Lemma 81.14.2
- Lemma 81.14.3
- Lemma 81.14.4
- Lemma 81.14.5
- Lemma 81.14.6
- Lemma 81.14.7
- Lemma 81.14.8
- Lemma 81.14.9
- Lemma 81.14.10
- Example 81.14.11
- Remark 81.14.12: Warning
- Lemma 81.14.13
- Lemma 81.14.14
Section 81.15: Types of formal algebraic spaces
- Lemma 81.15.1
- Lemma 81.15.2
  - Equation 81.15.2.1
  - Equation 81.15.2.2
- Lemma 81.15.3
- Lemma 81.15.4
- Definition 81.15.5
- Lemma 81.15.6
- Remark 81.15.7: Warning
Section 81.16: Morphisms and continuous ring maps
- Lemma 81.16.1
- Situation 81.16.2
  - Equation 81.16.2.1
- Lemma 81.16.3
- Remark 81.16.4: Variant for adic-star
- Remark 81.16.5: Variant for Noetherian
- Lemma 81.16.6
- Lemma 81.16.7
Section 81.17: Adic morphisms
- Definition 81.17.1
Section 81.18: Morphisms of finite type
- Definition 81.18.1
- Lemma 81.18.2
- Lemma 81.18.3
- Lemma 81.18.4
- Lemma 81.18.5
- Lemma 81.18.6
- Example 81.18.7
- Lemma 81.18.8
- Lemma 81.18.9
- Lemma 81.18.10
Section 81.19: Monomorphisms
- Definition 81.19.1
Section 81.20: Closed immersions
- Definition 81.20.1
- Lemma 81.20.2
- Example 81.20.3
Section 81.21: Restricted power series
- Remark 81.21.1: Universal property restricted power series reference
- Remark 81.21.2
Section 81.22: Algebras topologically of finite type
- Definition 81.22.1
- Lemma 81.22.2
- Lemma 81.22.3
- Remark 81.22.4
- Lemma 81.22.5
- Lemma 81.22.6
- Lemma 81.22.7
- Lemma 81.22.8
- Lemma 81.22.9
Section 81.23: Separation axioms for morphisms
- Definition 81.23.1
- Lemma 81.23.2
- Lemma 81.23.3
- Lemma 81.23.4
- Lemma 81.23.5
Section 81.24: Proper morphisms
- Definition 81.24.1
- Lemma 81.24.2
Section 81.25: Formal algebraic spaces and fpqc coverings
- Lemma 81.25.1 slogan
Section 81.26: Maps out of affine formal schemes
- Lemma 81.26.1
- Lemma 81.26.2
- Lemma 81.26.3
Section 81.27: The small étale site of a formal algebraic space
- Definition 81.27.1
  - Equation 81.27.1.1
- Lemma 81.27.2
- Lemma 81.27.3
- Lemma 81.27.4
- Lemma 81.27.5
- Lemma 81.27.6
- Lemma 81.27.7
Section 81.28: The structure sheaf
- Lemma 81.28.1
- Remark 81.28.2
- Lemma 81.28.3
- Remark 81.28.4