The Stacks project

Lemma 28.27.1. Let $X$ be a scheme. Then $X$ is quasi-affine if and only if $\mathcal{O}_ X$ is ample.

Proof. Suppose that $X$ is quasi-affine. Set $A = \Gamma (X, \mathcal{O}_ X)$. Consider the open immersion

\[ j : X \longrightarrow \mathop{\mathrm{Spec}}(A) \]

from Lemma 28.18.4. Note that $\mathop{\mathrm{Spec}}(A) = \text{Proj}(A[T])$, see Constructions, Example 27.8.14. Hence we can apply Lemma 28.26.12 to deduce that $\mathcal{O}_ X$ is ample.

Suppose that $\mathcal{O}_ X$ is ample. Note that $\Gamma _*(X, \mathcal{O}_ X) \cong A[T]$ as graded rings. Hence the result follows from Lemmas 28.26.11 and 28.18.4 taking into account that $\mathop{\mathrm{Spec}}(A) = \text{Proj}(A[T])$ for any ring $A$ as seen above. $\square$

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