The Stacks project

Lemma 38.34.4. Let $X$ be an affine scheme. Let $\{ X_ i \to X\} _{i \in I}$ be an h covering. Then there exists a surjective proper morphism

\[ Y \longrightarrow X \]

of finite presentation (!) and a finite affine open covering $Y = \bigcup _{j = 1, \ldots , m} Y_ j$ such that $\{ Y_ j \to X\} _{j = 1, \ldots , m}$ refines $\{ X_ i \to X\} _{i \in I}$.

Proof. By assumption there exists a proper surjective morphism $Y \to X$ and a finite affine open covering $Y = \bigcup _{j = 1, \ldots , m} Y_ j$ such that $\{ Y_ j \to X\} _{j = 1, \ldots , m}$ refines $\{ X_ i \to X\} _{i \in I}$. This means that for each $j$ there is an index $i_ j \in I$ and a morphism $h_ j : Y_ j \to X_{i_ j}$ over $X$. See Definition 38.34.2 and Topologies, Definition 34.8.4. The problem is that we don't know that $Y \to X$ is of finite presentation. By Limits, Lemma 32.13.2 we can write

\[ Y = \mathop{\mathrm{lim}}\nolimits Y_\lambda \]

as a directed limit of schemes $Y_\lambda $ proper and of finite presentation over $X$ such that the morphisms $Y \to Y_\lambda $ and the the transition morphisms are closed immersions. Observe that each $Y_\lambda \to X$ is surjective. By Limits, Lemma 32.4.11 we can find a $\lambda $ and quasi-compact opens $Y_{\lambda , j} \subset Y_\lambda $, $j = 1, \ldots , m$ covering $Y_\lambda $ and restricting to $Y_ j$ in $Y$. Then $Y_ j = \mathop{\mathrm{lim}}\nolimits Y_{\lambda , j}$. After increasing $\lambda $ we may assume $Y_{\lambda , j}$ is affine for all $j$, see Limits, Lemma 32.4.13. Finally, since $X_ i \to X$ is locally of finite presentation we can use the functorial characterization of morphisms which are locally of finite presentation (Limits, Proposition 32.6.1) to find a $\lambda $ such that for each $j$ there is a morphism $h_{\lambda , j} : Y_{\lambda , j} \to X_{i_ j}$ whose restriction to $Y_ j$ is the morphism $h_ j$ chosen above. Thus $\{ Y_{\lambda , j} \to X\} $ refines $\{ X_ i \to X\} $ and the proof is complete. $\square$


Comments (2)

Comment #4559 by on

The wording "direct limit" just under the display in the proof might be misleading since other books sometimes use "direct limit" to mean "colimit" or "filtered colimit", whereas here a limit is intended. Maybe change it to "directed limit", as in the lemma being referred to?


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