The Stacks project

Lemma 37.70.6. Let $X$ be a Noetherian scheme of dimension $\infty > d \geq 0$. Then the affine stratification number of $X$ is at most $d$.

Proof. By induction on $d$. If $d = 0$, then $X$ is affine, see Properties, Lemma 28.10.5. Assume $d > 0$. Let $\eta _1, \ldots , \eta _ n$ be the generic points of the irreducible components of $X$ (Properties, Lemma 28.5.7). We can cover $X$ by affine opens containing $\eta _1, \ldots , \eta _ n$, see Properties, Lemma 28.29.4. Since $X$ is quasi-compact we can find a finite affine open covering $X = \bigcup _{j = 1, \ldots , m} U_ j$ with $\eta _1, \ldots , \eta _ n \in U_ j$ for all $j = 1, \ldots , m$. Choose an affine open $U \subset U_1 \cap \ldots \cap U_ m$ containing $\eta _1, \ldots , \eta _ n$ (possible by the lemma already quoted). Then the morphism $U \to X$ is affine because $U \to U_ j$ is affine for all $j$, see Morphisms, Lemma 29.11.3. Let $Z = X \setminus U$. By construction $\dim (Z) < \dim (X)$. By induction hypothesis we can find an affine stratification $Z = \bigcup _{i \in \{ 0, \ldots , n\} } Z_ i$ of $Z$ with $n \leq \dim (Z)$. Setting $U = X_{n + 1}$ and $X_ i = Z_ i$ for $i \leq n$ we conclude. $\square$


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