Lemma 32.11.6. Let $X$ be a scheme. Let $\mathcal{L}$ be an ample invertible sheaf on $X$. Assume we have morphisms of schemes

\[ \mathop{\mathrm{Spec}}(k) \leftarrow \mathop{\mathrm{Spec}}(A) \to W \subset X \]

where $k$ is a field, $A$ is an integral $k$-algebra, $W$ is open in $X$. Then there exists an $n > 0$ and a section $s \in \Gamma (X, \mathcal{L}^{\otimes n})$ such that $X_ s$ is affine, $X_ s \subset W$, and $\mathop{\mathrm{Spec}}(A) \to W$ factors through $X_ s$

**Proof.**
Since $\mathop{\mathrm{Spec}}(A)$ is quasi-compact, we may replace $W$ by a quasi-compact open still containing the image of $\mathop{\mathrm{Spec}}(A) \to X$. Recall that $X$ is quasi-separated and quasi-compact by dint of having an ample invertible sheaf, see Properties, Definition 28.26.1 and Lemma 28.26.7. By Proposition 32.5.4 we can write $X = \mathop{\mathrm{lim}}\nolimits X_ i$ as a limit of a directed system of schemes of finite type over $\mathbf{Z}$ with affine transition morphisms. For some $i$ the ample invertible sheaf $\mathcal{L}$ on $X$ descends to an ample invertible sheaf $\mathcal{L}_ i$ on $X_ i$ and the open $W$ is the inverse image of a quasi-compact open $W_ i \subset X_ i$, see Lemmas 32.4.15, 32.10.3, and 32.4.11. We may replace $X, W, \mathcal{L}$ by $X_ i, W_ i, \mathcal{L}_ i$ and assume $X$ is of finite presentation over $\mathbf{Z}$. Write $A = \mathop{\mathrm{colim}}\nolimits A_ j$ as the colimit of its finite $k$-subalgebras. Then for some $j$ the morphism $\mathop{\mathrm{Spec}}(A) \to X$ factors through a morphism $\mathop{\mathrm{Spec}}(A_ j) \to X$, see Proposition 32.6.1. Since $\mathop{\mathrm{Spec}}(A_ j)$ is finite this reduces the lemma to Properties, Lemma 28.29.6.
$\square$

## Comments (0)