Chapter 66 (03BO): Properties of Algebraic Spaces

Section 66.1: Introduction

Section 66.2: Conventions

Section 66.3: Separation axioms

Definition 66.3.1
Lemma 66.3.2
Lemma 66.3.3
Lemma 66.3.4

Section 66.4: Points of algebraic spaces

Section 66.5: Quasi-compact spaces

Definition 66.5.1
Lemma 66.5.2
Lemma 66.5.3
Example 66.5.4
Lemma 66.5.5

Section 66.6: Special coverings

Lemma 66.6.1
Lemma 66.6.2
Lemma 66.6.3
Lemma 66.6.4
Lemma 66.6.5
Lemma 66.6.6
Lemma 66.6.7

Section 66.7: Properties of Spaces defined by properties of schemes

Lemma 66.7.1
Definition 66.7.2
Remark 66.7.3
Lemma 66.7.4
Definition 66.7.5
Remark 66.7.6

Section 66.8: Constructible sets

Lemma 66.8.1
Definition 66.8.2

Section 66.9: Dimension at a point

Definition 66.9.1
Definition 66.9.2

Section 66.10: Dimension of local rings

Lemma 66.10.1
Definition 66.10.2
Lemma 66.10.3

Section 66.11: Generic points

Lemma 66.11.1
Lemma 66.11.2

Section 66.12: Reduced spaces

Lemma 66.12.1
Lemma 66.12.2
Lemma 66.12.3
Lemma 66.12.4
Definition 66.12.5

Section 66.13: The schematic locus

Lemma 66.13.1
Lemma 66.13.2
Proposition 66.13.3

Section 66.14: Obtaining a scheme

Proposition 66.14.1
Lemma 66.14.2
Lemma 66.14.3

Section 66.15: Points on quasi-separated spaces

Lemma 66.15.1
Lemma 66.15.2
Lemma 66.15.3
Remark 66.15.4

Section 66.16: Étale morphisms of algebraic spaces

Section 66.17: Spaces and fpqc coverings

Proposition 66.17.1: Gabber

Section 66.18: The étale site of an algebraic space

Section 66.19: Points of the small étale site

Section 66.20: Supports of abelian sheaves

Lemma 66.20.1
Lemma 66.20.2
Definition 66.20.3
Lemma 66.20.4
Lemma 66.20.5

Section 66.21: The structure sheaf of an algebraic space

Lemma 66.21.1
Definition 66.21.2
Lemma 66.21.3
Lemma 66.21.4

Section 66.22: Stalks of the structure sheaf

Lemma 66.22.1
Definition 66.22.2
Lemma 66.22.3
Lemma 66.22.4
Lemma 66.22.5
Lemma 66.22.6

Section 66.23: Local irreducibility

Lemma 66.23.1
Definition 66.23.2
Lemma 66.23.3
Definition 66.23.4

Section 66.24: Noetherian spaces

Definition 66.24.1
Lemma 66.24.2
Lemma 66.24.3
Lemma 66.24.4

Section 66.25: Regular algebraic spaces

Lemma 66.25.1
Definition 66.25.2
Lemma 66.25.3
Lemma 66.25.4

Section 66.26: Sheaves of modules on algebraic spaces

Equation 66.26.0.1
Lemma 66.26.1
- Equation 66.26.1.1
Lemma 66.26.2
Lemma 66.26.3

Section 66.27: Étale localization

Equation 66.27.0.1
Equation 66.27.0.2
Equation 66.27.0.3
Equation 66.27.0.4
Equation 66.27.0.5
Equation 66.27.0.6
Lemma 66.27.1
Lemma 66.27.2
Lemma 66.27.3

Section 66.28: Recovering morphisms

Lemma 66.28.1
Lemma 66.28.2
Lemma 66.28.3
Theorem 66.28.4
Lemma 66.28.5

Section 66.29: Quasi-coherent sheaves on algebraic spaces

Definition 66.29.1
Lemma 66.29.2
Lemma 66.29.3
Lemma 66.29.4
Lemma 66.29.5
Lemma 66.29.6
Lemma 66.29.7

Section 66.30: Properties of modules

Section 66.31: Locally projective modules

Lemma 66.31.1
Definition 66.31.2
Lemma 66.31.3

Section 66.32: Quasi-coherent sheaves and presentations

Proposition 66.32.1
Proposition 66.32.2

Section 66.33: Morphisms towards schemes

Lemma 66.33.1

Section 66.34: Quotients by free actions

Lemma 66.34.1

The Stacks project

66 Properties of Algebraic Spaces