# Definition:Separated Morphism of Schemes

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## 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.

This article is complete as far as it goes, but it could do with expansion.In particular: If these are truly equivalent definitions, and can be found in some source work as actual definitions, then they are to be entered as actual definition pages according to the $\mathsf{Pr} \infty \mathsf{fWiki}$ convention.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Expand}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Also see

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

## Sources

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