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.

## 32.11 Characterizing affine schemes

If $f : X \to S$ is a surjective integral morphism of schemes such that $X$ is an affine scheme then $S$ is affine too. See [A.2, Conrad-Nagata]. Our proof relies on the Noetherian case which we stated and proved in Cohomology of Schemes, Lemma 30.13.3. See also [II 6.7.1, EGA].

**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. $\square$

Proposition 32.11.2. Let $f : X \to S$ be a morphism of schemes. Assume that $f$ is surjective and integral, 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 32.7.2 we can write $X = \mathop{\mathrm{lim}}\nolimits _ i X_ i$ with $X_ i \to S$ finite. By Lemma 32.4.13 we see that for $i$ sufficiently large the scheme $X_ i$ is affine. Moreover, since $X \to S$ factors through each $X_ i$ we see that $X_ i \to S$ is surjective. Hence we conclude that $S$ is affine by Lemma 32.11.1. $\square$

Lemma 32.11.3. Let $X$ be a scheme which is set theoretically the union of finitely many affine closed subschemes. Then $X$ is affine.

**Proof.**
Let $Z_ i \subset X$, $i = 1, \ldots , n$ be affine closed subschemes such that $X = \bigcup Z_ i$ set theoretically. Then $\coprod Z_ i \to X$ is surjective and integral with affine source. Hence $X$ is affine by Proposition 32.11.2.
$\square$

Lemma 32.11.4. Let $i : Z \to X$ be a closed immersion of schemes inducing a homeomorphism of underlying topological spaces. Let $\mathcal{L}$ be an invertible sheaf on $X$. Then $i^*\mathcal{L}$ is ample on $Z$, if and only if $\mathcal{L}$ is ample on $X$.

**Proof.**
If $\mathcal{L}$ is ample, then $i^*\mathcal{L}$ is ample for example by Morphisms, Lemma 29.37.7. Assume $i^*\mathcal{L}$ is ample. Then $Z$ is quasi-compact (Properties, Definition 28.26.1) and separated (Properties, Lemma 28.26.8). Since $i$ is surjective, we see that $X$ is quasi-compact. Since $i$ is universally closed and surjective, we see that $X$ is separated (Morphisms, Lemma 29.41.11).

By Proposition 32.5.4 we can write $X = \mathop{\mathrm{lim}}\nolimits X_ i$ as a directed limit of finite type schemes over $\mathbf{Z}$ with affine transition morphisms. We can find an $i$ and an invertible sheaf $\mathcal{L}_ i$ on $X_ i$ whose pullback to $X$ is isomorphic to $\mathcal{L}$, see Lemma 32.10.2.

For each $i$ let $Z_ i \subset X_ i$ be the scheme theoretic image of the morphism $Z \to X$. If $\mathop{\mathrm{Spec}}(A_ i) \subset X_ i$ is an affine open subscheme with inverse image of $\mathop{\mathrm{Spec}}(A)$ in $X$ and if $Z \cap \mathop{\mathrm{Spec}}(A)$ is defined by the ideal $I \subset A$, then $Z_ i \cap \mathop{\mathrm{Spec}}(A_ i)$ is defined by the ideal $I_ i \subset A_ i$ which is the inverse image of $I$ in $A_ i$ under the ring map $A_ i \to A$, see Morphisms, Example 29.6.4. Since $\mathop{\mathrm{colim}}\nolimits A_ i/I_ i = A/I$ it follows that $\mathop{\mathrm{lim}}\nolimits Z_ i = Z$. By Lemma 32.4.15 we see that $\mathcal{L}_ i|_{Z_ i}$ is ample for some $i$. Since $Z$ and hence $X$ maps into $Z_ i$ set theoretically, we see that $X_{i'} \to X_ i$ maps into $Z_ i$ set theoretically for some $i' \geq i$, see Lemma 32.4.10. (Observe that since $X_ i$ is Noetherian, every closed subset of $X_ i$ is constructible.) Let $T \subset X_{i'}$ be the scheme theoretic inverse image of $Z_ i$ in $X_{i'}$. Observe that $\mathcal{L}_{i'}|_ T$ is the pullback of $\mathcal{L}_ i|_{Z_ i}$ and hence ample by Morphisms, Lemma 29.37.7 and the fact that $T \to Z_ i$ is an affine morphism. Thus we see that $\mathcal{L}_{i'}$ is ample on $X_{i'}$ by Cohomology of Schemes, Lemma 30.17.5. Pulling back to $X$ (using the same lemma as above) we find that $\mathcal{L}$ is ample. $\square$

Lemma 32.11.5. Let $i : Z \to X$ be a closed immersion of schemes inducing a homeomorphism of underlying topological spaces. Then $X$ is quasi-affine if and only if $Z$ is quasi-affine.

**Proof.**
Recall that a scheme is quasi-affine if and only if the structure sheaf is ample, see Properties, Lemma 28.27.1. Hence if $Z$ is quasi-affine, then $\mathcal{O}_ Z$ is ample, hence $\mathcal{O}_ X$ is ample by Lemma 32.11.4, hence $X$ is quasi-affine. A proof of the converse, which can also be seen in an elementary way, is gotten by reading the argument just given backwards.
$\square$

The following lemma does not really belong in this section.

Lemma 32.11.6. Let $X$ be a scheme. Let $\mathcal{L}$ be an ample invertible sheaf on $X$. Assume we have morphisms of schemes

where $k$ is a field, $A$ is an integral $k$-algebra, $W$ is open in $X$. Then there exists an $n > 0$ and a section $s \in \Gamma (X, \mathcal{L}^{\otimes n})$ such that $X_ s$ is affine, $X_ s \subset W$, and $\mathop{\mathrm{Spec}}(A) \to W$ factors through $X_ s$

**Proof.**
Since $\mathop{\mathrm{Spec}}(A)$ is quasi-compact, we may replace $W$ by a quasi-compact open still containing the image of $\mathop{\mathrm{Spec}}(A) \to X$. Recall that $X$ is quasi-separated and quasi-compact by dint of having an ample invertible sheaf, see Properties, Definition 28.26.1 and Lemma 28.26.7. By Proposition 32.5.4 we can write $X = \mathop{\mathrm{lim}}\nolimits X_ i$ as a limit of a directed system of schemes of finite type over $\mathbf{Z}$ with affine transition morphisms. For some $i$ the ample invertible sheaf $\mathcal{L}$ on $X$ descends to an ample invertible sheaf $\mathcal{L}_ i$ on $X_ i$ and the open $W$ is the inverse image of a quasi-compact open $W_ i \subset X_ i$, see Lemmas 32.4.15, 32.10.3, and 32.4.11. We may replace $X, W, \mathcal{L}$ by $X_ i, W_ i, \mathcal{L}_ i$ and assume $X$ is of finite presentation over $\mathbf{Z}$. Write $A = \mathop{\mathrm{colim}}\nolimits A_ j$ as the colimit of its finite $k$-subalgebras. Then for some $j$ the morphism $\mathop{\mathrm{Spec}}(A) \to X$ factors through a morphism $\mathop{\mathrm{Spec}}(A_ j) \to X$, see Proposition 32.6.1. Since $\mathop{\mathrm{Spec}}(A_ j)$ is finite this reduces the lemma to Properties, Lemma 28.29.6.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)