Definition:Separated Morphism of Schemes

Definition

Let $\struct {X, \OO_X}$ and $\struct {Y, \OO_Y}$ be schemes.

Let $f : \struct {X, \OO_X} \to \struct {Y, \OO_Y}$ be a morphism of schemes.

$f$ is separated if and only if $f$ the diagonal morphism $\Delta_f : X \times_Y X \to Y$ is a closed immersion.

Also defined as

Equivalent definitions include:

• $f$ is separated if and only if $\Delta_f$ is closed.
• $f$ is separated if and only if $\Img {\Delta_f}$ is closed.

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Also see

• Equivalence of Definitions of Separated Morphism of Schemes

• Definition:Quasi-Separated Morphism of Schemes
• Definition:Proper Morphism of Schemes
• Separated Morphism is Quasi-Separated

Sources

• 1977: Robin Hartshorne: Algebraic Geometry $\S \text{II}.4$