The Stacks project

Lemma 37.71.3. Let $X$ be a scheme. The following are equivalent

  1. $X$ has a finite affine stratification, and

  2. $X$ is quasi-compact and quasi-separated.

Proof. Let $X = \bigcup X_ i$ be a finite affine stratification. Since each $X_ i$ is affine hence quasi-compact, we conclude that $X$ is quasi-compact. Let $U, V \subset X$ be affine open. Then $U \cap X_ i$ and $V \cap X_ i$ are affine open in $X_ i$ since $X_ i \to X$ is an affine morphism. Hence $U \cap V \cap X_ i$ is an affine open of the affine scheme $X_ i$ (see Schemes, Lemma 26.21.7 for example). Therefore $U \cap V = \coprod U \cap V \cap X_ i$ is quasi-compact as a finite union of affine strata. We conclude that $X$ is quasi-separated by Schemes, Lemma 26.21.6.

Assume $X$ is quasi-compact and quasi-separated. We may use the induction principle of Cohomology of Schemes, Lemma 30.4.1 to prove the assertion that $X$ has a finite affine stratification. If $X$ is empty, then it has an empty affine stratification. If $X$ is nonempty affine then it has an affine stratification with one stratum. Next, asssume $X = U \cup V$ where $U$ is quasi-compact open, $V$ is affine open, and we have a finite affine stratifications $U = \bigcup _{i \in I} U_ i$ and $U \cap V = \coprod _{j \in J} W_ j$. Denote $Z = X \setminus V$ and $Z' = X \setminus U$. Note that $Z$ is closed in $U$ and $Z'$ is closed in $V$. Observe that $U_ i \cap Z$ and $U_ i \cap W_ j = U_ i \times _ U W_ j$ are affine schemes affine over $U$. (Hints: use that $U_ i \times _ U W_ j \to W_ j$ is affine as a base change of $U_ i \to U$, hence $U_ i \cap W_ j$ is affine, hence $U_ i \cap W_ j \to U_ i$ is affine, hence $U_ i \cap W_ j \to U$ is affine.) It follows that

\[ U = \coprod \nolimits _{i \in I} (U_ i \cap Z) \amalg \coprod \nolimits _{(i, j) \in I \times J} (U_ i \cap W_ j) \]

is a finite affine stratification with partial ordering on $I \amalg I \times J$ given by $i' \leq (i, j) \Leftrightarrow i' \leq i$ and $(i', j') \leq (i, j) \Leftrightarrow i' \leq i$ and $j' \leq j$. Observe that $(U_ i \cap Z) \times _ X V = \emptyset $ and $(U_ i \cap W_ j) \times _ X V = U_ i \cap W_ j$ are affine. Hence the morphisms $U_ i \cap Z \to X$ and $U_ i \cap W_ j \to X$ are affine because we can check affineness of a morphism locally on the target (Morphisms, Lemma 29.11.3) and we have affineness over both $U$ and $V$. To finish the proof we take the stratification above and we add one additional stratum, namely $Z'$, whose index we add as a minimal element to the partially ordered set. $\square$

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