The Stacks project

Lemma 31.4.16. Let $S$ be a scheme. Let $X = \mathop{\mathrm{lim}}\nolimits X_ i$ be a directed limit of schemes over $S$ with affine transition morphisms. Let $Y \to X$ be a morphism of schemes over $S$.

  1. If $Y \to X$ is a closed immersion, $X_ i$ quasi-compact, and $Y$ locally of finite type over $S$, then $Y \to X_ i$ is a closed immersion for $i$ large enough.

  2. If $Y \to X$ is an immersion, $X_ i$ quasi-separated, $Y \to S$ locally of finite type, and $Y$ quasi-compact, then $Y \to X_ i$ is an immersion for $i$ large enough.

  3. If $Y \to X$ is an isomorphism, $X_ i$ quasi-compact, $X_ i \to S$ locally of finite type, the transition morphisms $X_{i'} \to X_ i$ are closed immersions, and $Y \to S$ is locally of finite presentation, then $Y \to X_ i$ is an isomorphism for $i$ large enough.

Proof. Proof of (1). Choose $0 \in I$ and a finite affine open covering $X_0 = U_{0, 1} \cup \ldots \cup U_{0, m}$ with the property that $U_{0, j}$ maps into an affine open $W_ j \subset S$. Let $V_ j \subset Y$, resp. $U_{i, j} \subset X_ i$, $i \geq 0$, resp. $U_ j \subset X$ be the inverse image of $U_{0, j}$. It suffices to prove that $V_ j \to U_{i, j}$ is a closed immersion for $i$ sufficiently large and we know that $V_ j \to U_ j$ is a closed immersion. Thus we reduce to the following algebra fact: If $A = \mathop{\mathrm{colim}}\nolimits A_ i$ is a directed colimit of $R$-algebras, $A \to B$ is a surjection of $R$-algebras, and $B$ is a finitely generated $R$-algebra, then $A_ i \to B$ is surjective for $i$ sufficiently large.

Proof of (2). Choose $0 \in I$. Choose a quasi-compact open $X'_0 \subset X_0$ such that $Y \to X_0$ factors through $X'_0$. After replacing $X_ i$ by the inverse image of $X'_0$ for $i \geq 0$ we may assume all $X_ i'$ are quasi-compact and quasi-separated. Let $U \subset X$ be a quasi-compact open such that $Y \to X$ factors through a closed immersion $Y \to U$ ($U$ exists as $Y$ is quasi-compact). By Lemma 31.4.11 we may assume that $U = \mathop{\mathrm{lim}}\nolimits U_ i$ with $U_ i \subset X_ i$ quasi-compact open. By part (1) we see that $Y \to U_ i$ is a closed immersion for some $i$. Thus (2) holds.

Proof of (3). Working affine locally on $X_0$ for some $0 \in I$ as in the proof of (1) we reduce to the following algebra fact: If $A = \mathop{\mathrm{lim}}\nolimits A_ i$ is a directed colimit of $R$-algebras with surjective transition maps and $A$ of finite presentation over $A_0$, then $A = A_ i$ for some $i$. Namely, write $A = A_0/(f_1, \ldots , f_ n)$. Pick $i$ such that $f_1, \ldots , f_ n$ map to zero under the surjective map $A_0 \to A_ i$. $\square$


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