The Stacks project

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

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

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

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

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

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

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. $\square$

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

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. $\square$

Proof. Let $A = \Gamma _*(X, \mathcal{L}) = \bigoplus _{d \geq 0} \Gamma (X, \mathcal{L}^{\otimes d})$. By Properties, Proposition 28.26.13 the set of affine opens $X_ a$ with $a \in A_{+}$ homogeneous forms a basis for the topology of $X$. Hence we can find finitely many such elements $a_0, \ldots , a_ n \in A_{+}$ such that

1. we have $X = \bigcup _{i = 0, \ldots , n} X_{a_ i}$,

2. each $X_{a_ i}$ is affine, and

3. each $X_{a_ i}$ maps into an affine open $V_ i \subset S$.

By Lemma 29.15.2 we see that the ring maps $\mathcal{O}_ S(V_ i) \to \mathcal{O}_ X(X_{a_ i})$ are of finite type. Hence we can find finitely many elements $f_{ij} \in \mathcal{O}_ X(X_{a_ i})$, $j = 1, \ldots , n_ i$ which generate $\mathcal{O}_ X(X_{a_ i})$ as an $\mathcal{O}_ S(V_ i)$-algebra. By Properties, Lemma 28.17.2 we may write each $f_{ij}$ as $a_{ij}/a_ i^{e_{ij}}$ for some $a_{ij} \in A_{+}$ homogeneous. Let $N$ be a positive integer which is a common multiple of all the degrees of the elements $a_ i$, $a_{ij}$. Consider the elements

By construction these generate the invertible sheaf $\mathcal{L}^{\otimes N}$ over $X$. Hence they give rise to a morphism

over $S$, see Constructions, Lemma 27.13.1 and Definition 27.13.2. Moreover, $j^*\mathcal{O}_{\mathbf{P}_ S}(1) = \mathcal{L}^{\otimes N}$. We name the homogeneous coordinates $T_0, \ldots , T_ n, T_{ij}$ instead of $T_0, \ldots , T_ m$. For $i = 0, \ldots , n$ we have $i^{-1}(D_{+}(T_ i)) = X_{a_ i}$. Moreover, pulling back the element $T_{ij}/T_ i$ via $j^\sharp $ we get the element $f_{ij} \in \mathcal{O}_ X(X_{a_ i})$. Hence the morphism $j$ restricted to $X_{a_ i}$ gives a closed immersion of $X_{a_ i}$ into the affine open $D_{+}(T_ i) \cap \mathbf{P}^ m_{V_ i}$ of $\mathbf{P}^ N_ S$. Hence we conclude that the morphism $j$ is an immersion. This implies the lemma holds for some $d$ and $n$ which is enough in virtually all applications.

This proves that for one $d_2 \geq 1$ (namely $d_2 = N$ above), some $m \geq 0$ there exists some immersion $j : X \to \mathbf{P}^ m_ S$ given by global sections $s'_0, \ldots , s'_ m \in \Gamma (X, \mathcal{L}^{\otimes d_2})$. By Properties, Proposition 28.26.13 we know there exists an integer $d_1$ such that $\mathcal{L}^{\otimes d}$ is globally generated for all $d \geq d_1$. Set $d_0 = d_1 + d_2$. We claim that the lemma holds with this value of $d_0$. Namely, given an integer $d \geq d_0$ we may choose $s''_1, \ldots , s''_ t \in \Gamma (X, \mathcal{L}^{\otimes d - d_2})$ which generate $\mathcal{L}^{\otimes d - d_2}$ over $X$. Set $k = (m + 1)t$ and denote $s_0, \ldots , s_ k$ the collection of sections $s'_\alpha s''_\beta $, $\alpha = 0, \ldots , m$, $\beta = 1, \ldots , t$. These generate $\mathcal{L}^{\otimes d}$ over $X$ and therefore define a morphism

such that $i^*\mathcal{O}_{\mathbf{P}^ n_ S}(1) \cong \mathcal{L}^{\otimes d}$. To see that $i$ is an immersion, observe that $i$ is the composition

where the first morphism is $(j, j')$ with $j'$ given by $s''_1, \ldots , s''_ t$ and the second morphism is the Segre embedding (Constructions, Lemma 27.13.6). Since $j$ is an immersion, so is $(j, j')$ (apply Lemma 29.3.1 to $X \to \mathbf{P}^ m_ S \times _ S \mathbf{P}^{t - 1}_ S \to \mathbf{P}^ m_ S$). Thus $i$ is a composition of immersions and hence an immersion (Schemes, Lemma 26.24.3). $\square$

Proof. The equivalence of (1) and (2) is Lemma 29.37.5. The implication (2) $\Rightarrow $ (6) is Lemma 29.39.3. Trivially (6) implies (5). As $\mathbf{P}^ n_ S$ is a projective bundle over $S$ (see Constructions, Lemma 27.21.5) we see that (5) implies (3) and (6) implies (4) from the definition of a relatively very ample sheaf. Trivially (4) implies (3). To finish we have to show that (3) implies (2) which follows from Lemma 29.38.2 and Lemma 29.37.2. $\square$

Proof. Trivially (3) implies (2). Lemma 29.38.2 guarantees that (2) implies (1) since a morphism of finite type is quasi-compact by definition. Assume that $\mathcal{L}$ is $f$-ample. Choose a finite affine open covering $S = V_1 \cup \ldots \cup V_ m$. Write $X_ i = f^{-1}(V_ i)$. By Lemma 29.39.4 above we see there exists a $d_0$ such that $\mathcal{L}^{\otimes d}$ is relatively very ample on $X_ i/V_ i$ for all $d \geq d_0$. Hence we conclude (1) implies (3) by Lemma 29.38.7. $\square$

Proof. We see that (1) implies (2) by taking an affine open covering of $S$ and applying Lemma 29.39.1 to each of the restrictions of $f$ and $\mathcal{L}$. We see that (2) implies (1) by Lemma 29.38.7. $\square$

Proof. We see that (1) implies (2) by taking an affine open covering of $S$ and applying Lemma 29.39.4 to each of the restrictions of $f$ and $\mathcal{L}$. We see that (2) implies (1) by Lemma 29.37.4. $\square$

Proof. By Lemma 29.39.6 we reduce to the case $S$ is affine. Combining Lemma 29.39.4 and Properties, Proposition 28.26.13 we can find an integer $d_0$ such that $\mathcal{N} \otimes \mathcal{L}^{\otimes d_0}$ is globally generated. Choose global sections $s_0, \ldots , s_ n$ of $\mathcal{N} \otimes \mathcal{L}^{\otimes d_0}$ which generate it. This determines a morphism $j : X \to \mathbf{P}^ n_ S$ over $S$. By Lemma 29.39.4 we can also pick an integer $d_1$ such that for all $d \geq d_1$ there exist sections $t_{d, 0}, \ldots , t_{d, n(d)}$ of $\mathcal{L}^{\otimes d}$ which generate it and define an immersion

over $S$. Then for $d \geq d_0 + d_1$ we can consider the morphism

This morphism is an immersion as it is the composition

where the first morphism is $(j, j_{d - d_0})$ and the second is the Segre embedding (Constructions, Lemma 27.13.6). Since $j$ is an immersion, so is $(j, j_{d - d_0})$ (apply Lemma 29.3.1). We have a composition of immersions and hence an immersion (Schemes, Lemma 26.24.3). $\square$