The Stacks project

A scheme, admitting a finite surjective map from an affine scheme, is affine.

Lemma 32.11.1. Let $f : X \to S$ be a morphism of schemes. Assume that $f$ is surjective and finite, and assume that $X$ is affine. Then $S$ is affine.

Proof. Since $f$ is surjective and $X$ is quasi-compact we see that $S$ is quasi-compact. Since $X$ is separated and $f$ is surjective and universally closed (Morphisms, Lemma 29.44.7), we see that $S$ is separated (Morphisms, Lemma 29.41.11).

By Lemma 32.9.8 we can write $X = \mathop{\mathrm{lim}}\nolimits _ a X_ a$ with $X_ a \to S$ finite and of finite presentation. By Lemma 32.4.13 we see that $X_ a$ is affine for some $a \in A$. Replacing $X$ by $X_ a$ we may assume that $X \to S$ is surjective, finite, of finite presentation and that $X$ is affine.

By Proposition 32.5.4 we may write $S = \mathop{\mathrm{lim}}\nolimits _{i \in I} S_ i$ as a directed limits of schemes of finite type over $\mathbf{Z}$. By Lemma 32.10.1 we can after shrinking $I$ assume there exist schemes $X_ i \to S_ i$ of finite presentation such that $X_{i'} = X_ i \times _ S S_{i'}$ for $i' \geq i$ and such that $X = \mathop{\mathrm{lim}}\nolimits _ i X_ i$. By Lemma 32.8.3 we may assume that $X_ i \to S_ i$ is finite for all $i \in I$ as well. By Lemma 32.4.13 once again we may assume that $X_ i$ is affine for all $i \in I$. Hence the result follows from the Noetherian case, see Cohomology of Schemes, Lemma 30.13.3. $\square$

Comments (5)

Comment #3024 by Brian Lawrence on

Suggested slogan: A scheme, admitting a finite surjective map from an affine scheme, is affine.

Comment #4056 by Laurent Moret-Bailly on

Typo in third paragraph of proof: "as a directed limit of schemes..."

Comment #4188 by BCnrd on

Why not mention the amusing consequence that if is an arbitrary scheme for which is affine then is affine (apply the above result with and the evident ? Or refer to where it is recorded if elsewhere (I don't see it anywhere in the ambient chapter).

Comment #4382 by on

@BCnrd: Yes, in some sense it is kind of silly that we didn't add this. On the other hand, we have the slighly stronger Lemma 32.11.3. So I think it is OK.

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