The Stacks project

Lemma 37.73.5. Let $X$ be a separated scheme which has an open covering by $n + 1$ affines. Then the affine stratification number of $X$ is at most $n$.

Proof. Say $X = U_0 \cup \ldots \cup U_ n$ is an affine open covering. Set

\[ X_ i = (U_ i \cup \ldots \cup U_ n) \setminus (U_{i + 1} \cup \ldots \cup U_ n) \]

Then $X_ i$ is affine as a closed subscheme of $U_ i$. The morphism $X_ i \to X$ is affine by Morphisms, Lemma 29.11.11. Finally, we have $\overline{X_ i} \subset X_ i \cup X_{i - 1} \cup \ldots X_0$. $\square$

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