The Stacks project

Remark 26.21.18. Let $P$ equal one of the following $4$ properties: “quasi-separated”, “separated”, “quasi-compact and quasi-separated”, or “quasi-compact and separated”. Then the following are true

1. Any affine scheme has $P$.

2. Given a morphism $f : X \to Y$ of scheme where $Y$ has $P$, then $f$ has $P$ if and only if $X$ has $P$.

3. If $X \to Y$ and $Z \to Y$ are morphisms of schemes and $X$, $Y$, $Z$ have $P$, then $X \times _ Y Z$ has $P$.

Statement (1) is clear. Let $f : X \to Y$ be as in (2). If $f$ has $P$, then so does $X$ by Lemmas 26.21.12 and 26.19.4 applied to $X \to Y \to \mathop{\mathrm{Spec}}(\mathbf{Z})$. If $X$ has $P$, then so does $f$ by Lemmas 26.21.13 and 26.21.14 applied to $X \to Y \to \mathop{\mathrm{Spec}}(\mathbf{Z})$. Let $X \to Y$ and $Z \to Y$ be as in (3). Then the projection $X \times _ Y Z \to Z$ has $P$ as a base change of the arrow $X \to Y$ which has $P$ by (2), see Lemmas 26.21.12 and 26.19.3. Hence $X \times _ Y Z$ has $P$ by (2).