The Stacks project

Lemma 68.17.3. 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 68.15.1 and 68.5.1. Then it follows that $X$ is affine from Proposition 68.16.7.

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 68.14.5. 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 68.17.1 and set $\mathcal{G} = f'_*\mathcal{O}_{Z'}$. We claim that $\mathcal{G}$ satisfies properties (3)(a) and (3)(b) of Lemma 68.14.5 which will finish the proof. Property (3)(a) we have seen in Lemma 68.17.1. 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 68.17.2 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 68.8.2. 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. $\square$


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