The Stacks project

Lemma 29.39.2. Let $\pi : X \to S$ be a morphism of schemes. Assume that $X$ is quasi-affine and that $\pi $ is locally of finite type. Then there exist $n \geq 0$ and an immersion $i : X \to \mathbf{A}^ n_ S$ over $S$.

Proof. Let $A = \Gamma (X, \mathcal{O}_ X)$. By assumption $X$ is quasi-compact and is identified with an open subscheme of $\mathop{\mathrm{Spec}}(A)$, see Properties, Lemma 28.18.4. Moreover, the set of opens $X_ f$, for those $f \in A$ such that $X_ f$ is affine, forms a basis for the topology of $X$, see the proof of Properties, Lemma 28.18.4. Hence we can find a finite number of $f_ j \in A$, $j = 1, \ldots , m$ such that $X = \bigcup X_{f_ j}$, and such that $\pi (X_{f_ j}) \subset V_ j$ for some affine open $V_ j \subset S$. By Lemma 29.15.2 the ring maps $\mathcal{O}(V_ j) \to \mathcal{O}(X_{f_ j}) = A_{f_ j}$ are of finite type. Thus we may choose $a_1, \ldots , a_ N \in A$ such that the elements $a_1, \ldots , a_ N, 1/f_ j$ generate $A_{f_ j}$ over $\mathcal{O}(V_ j)$ for each $j$. Take $n = m + N$ and let

\[ i : X \longrightarrow \mathbf{A}^ n_ S \]

be the morphism given by the global sections $f_1, \ldots , f_ m, a_1, \ldots , a_ N$ of the structure sheaf of $X$. Let $D(x_ j) \subset \mathbf{A}^ n_ S$ be the open subscheme where the $j$th coordinate function is nonzero. Then for $1 \leq j \leq m$ we have $i^{-1}(D(x_ j)) = X_{f_ j}$ and the induced morphism $X_{f_ j} \to D(x_ j)$ factors through the affine open $\mathop{\mathrm{Spec}}(\mathcal{O}(V_ j)[x_1, \ldots , x_ n, 1/x_ j])$ of $D(x_ j)$. Since the ring map $\mathcal{O}(V_ j)[x_1, \ldots , x_ n, 1/x_ j] \to A_{f_ j}$ is surjective by construction we conclude that $i^{-1}(D(x_ j)) \to D(x_ j)$ is an immersion as desired. $\square$

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