## Tag `03GI`

Chapter 25: Schemes > Section 25.21: Separation axioms

Lemma 25.21.15. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of schemes. If $g \circ f$ is quasi-compact and $g$ is quasi-separated then $f$ is quasi-compact.

Proof.This is true because $f$ equals the composition $(1, f) : X \to X \times_Z Y \to Y$. The first map is quasi-compact by Lemma 25.21.12 because it is a section of the quasi-separated morphism $X \times_Z Y \to X$ (a base change of $g$, see Lemma 25.21.13). The second map is quasi-compact as it is the base change of $g \circ f$, see Lemma 25.19.3. And compositions of quasi-compact morphisms are quasi-compact, see Lemma 25.19.4. $\square$

The code snippet corresponding to this tag is a part of the file `schemes.tex` and is located in lines 4200–4205 (see updates for more information).

```
\begin{lemma}
\label{lemma-quasi-compact-permanence}
Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of schemes.
If $g \circ f$ is quasi-compact and $g$ is quasi-separated
then $f$ is quasi-compact.
\end{lemma}
\begin{proof}
This is true because $f$ equals the composition
$(1, f) : X \to X \times_Z Y \to Y$. The first map
is quasi-compact by Lemma \ref{lemma-section-immersion}
because it is a section of the quasi-separated morphism $X \times_Z Y \to X$
(a base change of $g$, see Lemma \ref{lemma-separated-permanence}).
The second map is quasi-compact as it
is the base change of $g \circ f$, see
Lemma \ref{lemma-quasi-compact-preserved-base-change}.
And compositions of quasi-compact
morphisms are quasi-compact, see Lemma \ref{lemma-composition-quasi-compact}.
\end{proof}
```

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