Lemma 28.27.1 (01QE)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 28: Properties of Schemes
Section 28.27: Affine and quasi-affine schemes
Lemma 28.27.1 (cite)

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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. $\square$

Comments (2)

Comment #3564 by Fred Rohrer on September 03, 2018 at 15:29

Add that $A=\Gamma(X,\mathcal{O}_X)$ .

Comment #3688 by Johan on October 22, 2018 at 15:29

OK, thanks, and fixed here.

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