The Stacks project

69.17 Finite morphisms and affines

This section is the analogue of Cohomology of Schemes, Section 30.13.

Lemma 69.17.1. Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. Assume $f$ is finite, surjective and $X$ locally Noetherian. Let $i : Z \to X$ be a closed immersion. Denote $i' : Z' \to Y$ the inverse image of $Z$ (Morphisms of Spaces, Section 67.13) and $f' : Z' \to Z$ the induced morphism. Then $\mathcal{G} = f'_*\mathcal{O}_{Z'}$ is a coherent $\mathcal{O}_ Z$-module whose support is $Z$.

Proof. Observe that $f'$ is the base change of $f$ and hence is finite and surjective by Morphisms of Spaces, Lemmas 67.5.5 and 67.45.5. Note that $Y$, $Z$, and $Z'$ are locally Noetherian by Morphisms of Spaces, Lemma 67.23.5 (and the fact that closed immersions and finite morphisms are of finite type). By Lemma 69.12.9 we see that $\mathcal{G}$ is a coherent $\mathcal{O}_ Z$-module. The support of $\mathcal{G}$ is closed in $|Z|$, see Morphisms of Spaces, Lemma 67.15.2. Hence if the support of $\mathcal{G}$ is not equal to $|Z|$, then after replacing $X$ by an open subspace we may assume $\mathcal{G} = 0$ but $Z \not= \emptyset $. This would mean that $f'_*\mathcal{O}_{Z'} = 0$. In particular the section $1 \in \Gamma (Z', \mathcal{O}_{Z'}) = \Gamma (Z, f'_*\mathcal{O}_{Z'})$ would be zero which would imply $Z' = \emptyset $ is the empty algebraic space. This is impossible as $Z' \to Z$ is surjective. $\square$

Lemma 69.17.2. Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent sheaf on $Y$. Let $\mathcal{I}$ be a quasi-coherent sheaf of ideals on $X$. If $f$ is affine then $\mathcal{I}f_*\mathcal{F} = f_*(f^{-1}\mathcal{I}\mathcal{F})$ (with notation as explained in the proof).

Proof. The notation means the following. Since $f^{-1}$ is an exact functor we see that $f^{-1}\mathcal{I}$ is a sheaf of ideals of $f^{-1}\mathcal{O}_ X$. Via the map $f^\sharp : f^{-1}\mathcal{O}_ X \to \mathcal{O}_ Y$ on $Y_{\acute{e}tale}$ this acts on $\mathcal{F}$. Then $f^{-1}\mathcal{I}\mathcal{F}$ is the subsheaf generated by sums of local sections of the form $as$ where $a$ is a local section of $f^{-1}\mathcal{I}$ and $s$ is a local section of $\mathcal{F}$. It is a quasi-coherent $\mathcal{O}_ Y$-submodule of $\mathcal{F}$ because it is also the image of a natural map $f^*\mathcal{I} \otimes _{\mathcal{O}_ Y} \mathcal{F} \to \mathcal{F}$.

Having said this the proof is straightforward. Namely, the question is ├ętale local on $X$ and hence we may assume $X$ is an affine scheme. In this case the result is a consequence of the corresponding result for schemes, see Cohomology of Schemes, Lemma 30.13.2. $\square$

Lemma 69.17.3. Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. Assume

  1. $f$ finite,

  2. $f$ surjective,

  3. $Y$ affine, and

  4. $X$ Noetherian.

Then $X$ is affine.

Proof. We will prove that under the assumptions of the lemma for any coherent $\mathcal{O}_ X$-module $\mathcal{F}$ we have $H^1(X, \mathcal{F}) = 0$. This implies that $H^1(X, \mathcal{F}) = 0$ for every quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ by Lemmas 69.15.1 and 69.5.1. Then it follows that $X$ is affine from Proposition 69.16.7.

Let $\mathcal{P}$ be the property of coherent sheaves $\mathcal{F}$ on $X$ defined by the rule

\[ \mathcal{P}(\mathcal{F}) \Leftrightarrow H^1(X, \mathcal{F}) = 0. \]

We are going to apply Lemma 69.14.5. Thus we have to verify (1), (2) and (3) of that lemma for $\mathcal{P}$. Property (1) follows from the long exact cohomology sequence associated to a short exact sequence of sheaves. Property (2) follows since $H^1(X, -)$ is an additive functor. To see (3) let $i : Z \to X$ be a reduced closed subspace with $|Z|$ irreducible. Let $i' : Z' \to Y$ and $f' : Z' \to Z$ be as in Lemma 69.17.1 and set $\mathcal{G} = f'_*\mathcal{O}_{Z'}$. We claim that $\mathcal{G}$ satisfies properties (3)(a) and (3)(b) of Lemma 69.14.5 which will finish the proof. Property (3)(a) we have seen in Lemma 69.17.1. To see (3)(b) let $\mathcal{I}$ be a nonzero quasi-coherent sheaf of ideals on $Z$. Denote $\mathcal{I}' \subset \mathcal{O}_{Z'}$ the quasi-coherent ideal $(f')^{-1}\mathcal{I} \mathcal{O}_{Z'}$, i.e., the image of $(f')^*\mathcal{I} \to \mathcal{O}_{Z'}$. By Lemma 69.17.2 we have $f_*\mathcal{I}' = \mathcal{I} \mathcal{G}$. We claim the common value $\mathcal{G}' = \mathcal{I} \mathcal{G} = f'_*\mathcal{I}'$ satisfies the condition expressed in (3)(b). First, it is clear that the support of $\mathcal{G}/\mathcal{G}'$ is contained in the support of $\mathcal{O}_ Z/\mathcal{I}$ which is a proper subspace of $|Z|$ as $\mathcal{I}$ is a nonzero ideal sheaf on the reduced and irreducible algebraic space $Z$. The morphism $f'$ is affine, hence $R^1f'_*\mathcal{I}' = 0$ by Lemma 69.8.2. As $Z'$ is affine (as a closed subscheme of an affine scheme) we have $H^1(Z', \mathcal{I}') = 0$. Hence the Leray spectral sequence (in the form Cohomology on Sites, Lemma 21.14.6) implies that $H^1(Z, f'_*\mathcal{I}') = 0$. Since $i : Z \to X$ is affine we conclude that $R^1i_*f'_*\mathcal{I}' = 0$ hence $H^1(X, i_*f'_*\mathcal{I}') = 0$ by Leray again. In other words, we have $H^1(X, i_*\mathcal{G}') = 0$ as desired. $\square$

Comments (0)

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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 07VN. Beware of the difference between the letter 'O' and the digit '0'.