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 Bjorn Poonen on

Comment #4755 by Johan on