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

Also defined as
Equivalent definitions include:


 * $f$ is separated $\Delta_f$ is closed.
 * $f$ is separated $\Img {\Delta_f}$ is closed.

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