Lemma 30.13.3. Let f : Y \to X be a morphism of schemes. Assume
f finite,
f surjective,
Y affine, and
X Noetherian.
Then X is affine.
Lemma 30.13.3. Let f : Y \to X be a morphism of schemes. Assume
f finite,
f surjective,
Y affine, and
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 will in particular imply that H^1(X, \mathcal{I}) = 0 for every quasi-coherent sheaf of ideals of \mathcal{O}_ X. Then it follows that X is affine from either Lemma 30.3.1 or Lemma 30.3.2.
Let \mathcal{P} be the property of coherent sheaves \mathcal{F} on X defined by the rule
We are going to apply Lemma 30.12.8. 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 Z \subset X be an integral closed subscheme with generic point \xi . Let \mathcal{F} be a coherent sheaf on Y such that the support of f_*\mathcal{F} is equal to Z and (f_*\mathcal{F})_\xi is annihilated by \mathfrak m_\xi , see Lemma 30.13.1. We claim that taking \mathcal{G} = f_*\mathcal{F} works. We only have to verify part (3)(c) of Lemma 30.12.8. Hence assume that \mathcal{J} \subset \mathcal{O}_ X is a quasi-coherent sheaf of ideals such that \mathcal{J}_\xi = \mathcal{O}_{X, \xi }. A finite morphism is affine hence by Lemma 30.13.2 we see that \mathcal{J}\mathcal{G} = f_*(f^{-1}\mathcal{J}\mathcal{F}). Also, as pointed out in the proof of Lemma 30.13.2 the sheaf f^{-1}\mathcal{J}\mathcal{F} is a quasi-coherent \mathcal{O}_ Y-module. Since Y is affine we see that H^1(Y, f^{-1}\mathcal{J}\mathcal{F}) = 0, see Lemma 30.2.2. Since f is finite, hence affine, we see that
by Lemma 30.2.4. Hence the quasi-coherent subsheaf \mathcal{G}' = \mathcal{J}\mathcal{G} satisfies \mathcal{P}. This verifies property (3)(c) of Lemma 30.12.8 as desired. \square
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