The Stacks project

27.27 Affine and quasi-affine schemes

Lemma 27.27.1. 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 27.18.4. Note that $\mathop{\mathrm{Spec}}(A) = \text{Proj}(A[T])$, see Constructions, Example 26.8.14. Hence we can apply Lemma 27.26.12 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 27.26.11 and 27.18.4 taking into account that $\mathop{\mathrm{Spec}}(A) = \text{Proj}(A[T])$ for any ring $A$ as seen above. $\square$

Lemma 27.27.2. Let $X$ be a quasi-affine scheme. For any quasi-compact immersion $i : X' \to X$ the scheme $X'$ is quasi-affine.

Proof. This can be proved directly without making use of the material on ample invertible sheaves; we urge the reader to do this on a napkin. Since $X$ is quasi-affine, we have that $\mathcal{O}_ X$ is ample by Lemma 27.27.1. Then $\mathcal{O}_{X'}$ is ample by Lemma 27.26.14. Then $X'$ is quasi-affine by Lemma 27.27.1. $\square$

Lemma 27.27.3. Let $X$ be a scheme. Suppose that there exist finitely many elements $f_1, \ldots , f_ n \in \Gamma (X, \mathcal{O}_ X)$ such that

  1. each $X_{f_ i}$ is an affine open of $X$, and

  2. the ideal generated by $f_1, \ldots , f_ n$ in $\Gamma (X, \mathcal{O}_ X)$ is equal to the unit ideal.

Then $X$ is affine.

Proof. Assume we have $f_1, \ldots , f_ n$ as in the lemma. We may write $1 = \sum g_ i f_ i$ for some $g_ j \in \Gamma (X, \mathcal{O}_ X)$ and hence it is clear that $X = \bigcup X_{f_ i}$. (The $f_ i$'s cannot all vanish at a point.) Since each $X_{f_ i}$ is quasi-compact (being affine) it follows that $X$ is quasi-compact. Hence we see that $X$ is quasi-affine by Lemma 27.27.1 above. Consider the open immersion

\[ j : X \to \mathop{\mathrm{Spec}}(\Gamma (X, \mathcal{O}_ X)), \]

see Lemma 27.18.4. The inverse image of the standard open $D(f_ i)$ on the right hand side is equal to $X_{f_ i}$ on the left hand side and the morphism $j$ induces an isomorphism $X_{f_ i} \cong D(f_ i)$, see Lemma 27.18.3. Since the $f_ i$ generate the unit ideal we see that $\mathop{\mathrm{Spec}}(\Gamma (X, \mathcal{O}_ X)) = \bigcup _{i = 1, \ldots , n} D(f_ i)$. Thus $j$ is an isomorphism. $\square$

Comments (2)

Comment #2310 by Chia-Fu Yu on

In 27.27.1, one needs to assume that X is quasi-compact. In general quasi-affineness does not imply quasi-compactness. For example, let X=Spec A- m, where A=k[X_1,X_2,...] and m is the maximal idea m=(X_1,X_2,...).

Comment #2311 by on

Look at Definition 27.18.1 to see that quasi-affine does imply quasi-compactness (by definition).

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