Lemma 29.39.1. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible sheaf on $X$. Assume that

1. the invertible sheaf $\mathcal{L}$ is very ample on $X/S$,

2. the morphism $X \to S$ is of finite type, and

3. $S$ is affine.

Then there exist an $n \geq 0$ and an immersion $i : X \to \mathbf{P}^ n_ S$ over $S$ such that $\mathcal{L} \cong i^*\mathcal{O}_{\mathbf{P}^ n_ S}(1)$.

Proof. Assume (1), (2) and (3). Condition (3) means $S = \mathop{\mathrm{Spec}}(R)$ for some ring $R$. Condition (1) means by definition there exists a quasi-coherent $\mathcal{O}_ S$-module $\mathcal{E}$ and an immersion $\alpha : X \to \mathbf{P}(\mathcal{E})$ such that $\mathcal{L} = \alpha ^*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(1)$. Write $\mathcal{E} = \widetilde{M}$ for some $R$-module $M$. Thus we have

$\mathbf{P}(\mathcal{E}) = \text{Proj}(\text{Sym}_ R(M)).$

Since $\alpha$ is an immersion, and since the topology of $\text{Proj}(\text{Sym}_ R(M))$ is generated by the standard opens $D_{+}(f)$, $f \in \text{Sym}_ R^ d(M)$, $d \geq 1$, we can find for each $x \in X$ an $f \in \text{Sym}_ R^ d(M)$, $d \geq 1$, with $\alpha (x) \in D_{+}(f)$ such that

$\alpha |_{\alpha ^{-1}(D_{+}(f))} : \alpha ^{-1}(D_{+}(f)) \to D_{+}(f)$

is a closed immersion. Condition (2) implies $X$ is quasi-compact. Hence we can find a finite collection of elements $f_ j \in \text{Sym}_ R^{d_ j}(M)$, $d_ j \geq 1$ such that for each $f = f_ j$ the displayed map above is a closed immersion and such that $\alpha (X) \subset \bigcup D_{+}(f_ j)$. Write $U_ j = \alpha ^{-1}(D_{+}(f_ j))$. Note that $U_ j$ is affine as a closed subscheme of the affine scheme $D_{+}(f_ j)$. Write $U_ j = \mathop{\mathrm{Spec}}(A_ j)$. Condition (2) also implies that $A_ j$ is of finite type over $R$, see Lemma 29.15.2. Choose finitely many $x_{j, k} \in A_ j$ which generate $A_ j$ as a $R$-algebra. Since $\alpha |_{U_ j}$ is a closed immersion we see that $x_{j, k}$ is the image of an element

$f_{j, k}/f_ j^{e_{j, k}} \in \text{Sym}_ R(M)_{(f_ j)} = \Gamma (D_{+}(f_ j), \mathcal{O}_{\text{Proj}(\text{Sym}_ R(M))}).$

Finally, choose $n \geq 1$ and elements $y_0, \ldots , y_ n \in M$ such that each of the polynomials $f_ j, f_{j, k} \in \text{Sym}_ R(M)$ is a polynomial in the elements $y_ t$ with coefficients in $R$. Consider the graded ring map

$\psi : R[Y_0, \ldots , Y_ n] \longrightarrow \text{Sym}_ R(M), \quad Y_ i \longmapsto y_ i.$

Denote $F_ j$, $F_{j, k}$ the elements of $R[Y_0, \ldots , Y_ n]$ such that $\psi (F_ j) = f_ j$ and $\psi (F_{j, k}) = f_{j, k}$. By Constructions, Lemma 27.11.1 we obtain an open subscheme

$U(\psi ) \subset \text{Proj}(\text{Sym}_ R(M))$

and a morphism $r_\psi : U(\psi ) \to \mathbf{P}^ n_ R$. This morphism satisfies $r_\psi ^{-1}(D_{+}(F_ j)) = D_{+}(f_ j)$, and hence we see that $\alpha (X) \subset U(\psi )$. Moreover, it is clear that

$i = r_\psi \circ \alpha : X \longrightarrow \mathbf{P}^ n_ R$

is still an immersion since $i^\sharp (F_{j, k}/F_ j^{e_{j, k}}) = x_{j, k} \in A_ j = \Gamma (U_ j, \mathcal{O}_ X)$ by construction. Moreover, the morphism $r_\psi$ comes equipped with a map $\theta : r_\psi ^*\mathcal{O}_{\mathbf{P}^ n_ R}(1) \to \mathcal{O}_{\text{Proj}(\text{Sym}_ R(M))}(1)|_{U(\psi )}$ which is an isomorphism in this case (for construction $\theta$ see lemma cited above; some details omitted). Since the original map $\alpha$ was assumed to have the property that $\mathcal{L} = \alpha ^*\mathcal{O}_{\text{Proj}(\text{Sym}_ R(M))}(1)$ we win. $\square$

There are also:

• 2 comment(s) on Section 29.39: Ample and very ample sheaves relative to finite type morphisms

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 02NP. Beware of the difference between the letter 'O' and the digit '0'.