---
date: 2022-01-15 21:49
modification date: Friday 28th January 2022 18:34:19
title: separated
aliases: [quasiseparated, separated morphism, quasiseparated morphism, separated scheme]

---

Last modified date: <%+ tp.file.last_modified_date() %>

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- Tags: 
	- #AG/basics 
- Refs:
	- #todo/add-references
- Links:
	- [scheme](Unsorted/scheme.md)
	- [algebraic space](algebraic%20space.md)
	- [proper morphism](Unsorted/proper%20morphism.md)

---

# separated

Any morphism between affine schemes is separated.

We say a variety $X$ is separated if the image of the diagonal map $\Delta: X \rightarrow X \times X$ is Zariski closed in $X \times X$.

For instance, $\mathbb{C}^n$ is separated because the image of the diagonal in $\mathbb{C}^n \times \mathbb{C}^n=$ $\operatorname{Spec}\left(\mathbb{C}\left[x_1, \ldots, x_n, y_1, \ldots, y_n\right]\right)$ is the affine variety $\mathbf{V}\left(x_1-y_1, \ldots, x_n-y_n\right)$. Similarly, any affine variety is separated.



Idea: generalizes being **Hausdorff**. Clasically, $X\in \Top$ is Hausdorff iff $\Delta_X \subseteq X\cartpower{2}$ is a closed subspace.

- A *morphism* of schemes $f: X\to Y$ is **separated** if the relative diagonal $\Delta_{X/Y}: X\to X\fiberpower{Y}{2}$ is a [closed immersion](Unsorted/closed%20immersion.md).
	- Note that the [diagonal is always an immersion](https://stacks.math.columbia.edu/tag/01KJ), so the content is that the morphism is closed.
- A relative scheme $X\in\Sch\slice S$ is **separated** iff $\Delta_X \embeds X\fiberpower{S}{2}$ is a [closed immersion](closed%20immersion.md).
- A general *scheme* $Y\in \Sch$ is **separated** if the structure morphism $f: Y\to \spec \ZZ$ is separated, so $\Delta_{Y/ \spec \ZZ}: Y\to Y\fiberpower{\spec \ZZ}{2}$ is a [closed immersion](closed%20immersion.md).

# quasiseparated

Idea: intersections of two affines is [quasicompact](Unsorted/quasicompact.md), so a finite union of affines.

- A *morphism* of schemes $f: X\to Y$ is **quasiseparated** if the diagonal morphism $\Delta_Y: X\to X\fiberprod{Y}X$ is [quasicompact](quasicompact.md).
- A *scheme* $X\in \Sch$ is **quasiseparated** if the structure morphism $f: X\to \spec \ZZ$ is quasiseparated.
- An object $X$ in a [topos](topos.md) is quasiseparated iff or every pair of morphisms $U \rightarrow X \leftarrow V$, where $U$ and $V$ are quasi-compact, the fiber product $U \times_{X} V$ is also quasi-compact.

Note that separated implies quasiseparated.

# Exercises

- [ ] Show that $\AA^1\slice k$ with two origins is quasiseparated but not separated.
- [ ] Show that  [locally Noetherian](locally%20Noetherian) $\implies$ quasiseparated.