§4. Projective bundles. Ample sheaves

4.1. Definition of projective bundles

Definition.

Let $Y$ be a prescheme, $\mathcal{E}$ a quasi-coherent ${\mathcal{O}}_Y$-module, and ${\mathbb{S}}_{{\mathcal{O}}_Y}(\mathcal{E})$ the symmetric ${\mathcal{O}}_Y$-algebra of $\mathcal{E}$ (1.7.4), which is quasi-coherent (1.7.7). We call the projective bundle over $Y$ defined by $\mathcal{E}$, and denote by $\mathbb{P}(\mathcal{E})$, the $Y$-scheme $P=\mathrm{Proj}({\mathbb{S}}_{{\mathcal{O}}_Y}(\mathcal{E}))$. The ${\mathcal{O}}_P$-module ${\mathcal{O}}_P(1)$ is called the tautological sheaf on $P$.

Translator's note. EGA's 1961 term faisceau fondamental (literally "fundamental sheaf") is the modern tautological line bundle, or Serre line bundle, ${\mathcal{O}}_{\mathbb{P}(\mathcal{E})}(1)$. We render it as "tautological sheaf" throughout §4; the term is recorded in the ledger under §4.1.1.

When $Y$ is affine of ring $A$, we have $\mathcal{E}=E$ for some $A$-module $E$, and we then write $\mathbb{P}(E)$ in place of $\mathbb{P}(E)$.

When we take $\mathcal{E}={\mathcal{O}}_Y^n$, we write ${\mathbb{P}}_Y^{n-1}$ in place of $\mathbb{P}(\mathcal{E})$; if in addition $Y$ is affine of ring $A$, we also write ${\mathbb{P}}_A^{n-1}$ in place of ${\mathbb{P}}_Y^{n-1}$. Since ${\mathbb{S}}_{{\mathcal{O}}_Y}({\mathcal{O}}_Y)$ is canonically identified with ${\mathcal{O}}_Y[T]$ (1.7.4), ${\mathbb{P}}_Y^0$ is canonically identified with $Y$ (3.1.7); the example (2.4.3) is then nothing but ${\mathbb{P}}_K^1$.

(4.1.2)

Let $\mathcal{E}$, $\mathcal{F}$ be two quasi-coherent ${\mathcal{O}}_Y$-modules and $u:\mathcal{E}\to \mathcal{F}$ an ${\mathcal{O}}_Y$-homomorphism; there is canonically associated to it a homomorphism $\mathbb{S}(u):{\mathbb{S}}_{{\mathcal{O}}_Y}(\mathcal{E})\to {\mathbb{S}}_{{\mathcal{O}}_Y}(\mathcal{F})$ of graded ${\mathcal{O}}_Y$-algebras (1.7.4). If $u$ is surjective, then so is $\mathbb{S}(u)$, and therefore (3.6.2, (i)) $\mathrm{Proj}(\mathbb{S}(u))$ is a closed immersion $\mathbb{P}(\mathcal{F})\to \mathbb{P}(\mathcal{E})$, which we denote by $\mathbb{P}(u)$. We may therefore say that $\mathbb{P}(\mathcal{E})$ is a contravariant functor in $\mathcal{E}$, provided we restrict the morphisms of quasi-coherent ${\mathcal{O}}_Y$-modules to the surjective homomorphisms.

Still supposing $u$ surjective and setting $P=\mathbb{P}(\mathcal{E})$, $Q=\mathbb{P}(\mathcal{F})$, and $j=\mathbb{P}(u)$, we have, up to isomorphism,

  j*(𝒪_P(n)) = 𝒪_Q(n)            for all n ∈ ℤ,                            (4.1.2.1)

as follows from (3.6.3).

(4.1.3)

Now let $\psi :Y^{\prime }\to Y$ be a morphism and set ${\mathcal{E}}^{\prime }=\psi \ast (\mathcal{E})$; we then have ${\mathbb{S}}_{{\mathcal{O}}_{Y^{\prime }}}({\mathcal{E}}^{\prime })=\psi \ast ({\mathbb{S}}_{{\mathcal{O}}_Y}(\mathcal{E}))$ (1.7.5); hence (3.5.3),

  ℙ(ψ*(𝓔)) = ℙ(𝓔) ×_Y Y'                                                   (4.1.3.1)

up to canonical isomorphism. Furthermore, we evidently have

$$\psi \ast (({\mathbb{S}}_{{\mathcal{O}}_Y}(\mathcal{E}))(n))=({\mathbb{S}}_{{\mathcal{O}}_{Y^{\prime }}}({\mathcal{E}}^{\prime }))(n)$$

for all $n\in \mathbb{Z}$; setting $P=\mathbb{P}(\mathcal{E})$ and $P^{\prime }=\mathbb{P}({\mathcal{E}}^{\prime })$, we therefore have (3.5.4), up to isomorphism,

  𝒪_{P'}(n) = 𝒪_P(n) ⊗_Y 𝒪_{Y'}    for all n ∈ ℤ.                          (4.1.3.2)

Proposition.

Let $\mathcal{L}$ be an invertible ${\mathcal{O}}_Y$-module. For every quasi-coherent ${\mathcal{O}}_Y$-module $\mathcal{E}$, there exists a canonical $Y$-isomorphism $i_{\mathcal{L}}:\mathbb{P}(\mathcal{E})\stackrel{\sim }{\to }\mathbb{P}(\mathcal{E}\otimes \mathcal{L})$; furthermore, setting $P=\mathbb{P}(\mathcal{E})$ and $Q=\mathbb{P}(\mathcal{E}\otimes \mathcal{L})$, $i_{\mathcal{L}}\ast ({\mathcal{O}}_Q(n))$ is canonically isomorphic to ${\mathcal{O}}_P(n){\otimes }_Y{\mathcal{L}}^{\otimes n}$ for all $n\in \mathbb{Z}$.

Proof. Note first that if $A$ is a ring, $E$ an $A$-module, and $L$ a free monogenic $A$-module, one canonically defines a homomorphism of $A$-modules

  𝕊_n(E ⊗ L) → 𝕊_n(E) ⊗ L^{⊗n}

by sending $(x_1\otimes y_1)\cdots (x_n\otimes y_n)$ to

  (x_1 x_2 ⋯ x_n) ⊗ (y_1 ⊗ y_2 ⊗ ⋯ ⊗ y_n)         (x_i ∈ E, y_i ∈ L, for 1 ≤ i ≤ n).

One verifies immediately (by reducing to the case $L=A$) that this homomorphism is in fact an isomorphism. We conclude a canonical isomorphism of graded $A$-algebras ${\mathbb{S}}_A(E\otimes L)\stackrel{\sim }{\to }{\oplus }_{n\ge 0}{\mathbb{S}}_n(E)\otimes L^{\otimes n}$. Returning to the situation of (4.1.4), the preceding remarks allow us to define a canonical isomorphism of graded ${\mathcal{O}}_Y$-algebras

  𝕊_{𝒪_Y}(𝓔 ⊗_{𝒪_Y} ℒ) ⥲ ⊕_{n≥0} 𝕊_n(𝓔) ⊗_{𝒪_Y} ℒ^{⊗n}                    (4.1.4.1)

by defining this isomorphism as one of presheaves and using (1.7.4), (I, 1.3.9), and (I, 1.3.12). The proposition then follows from (3.1.8, (iii)) and (3.2.10).

(4.1.5)

With the hypotheses of (4.1.1), set $P=\mathbb{P}(\mathcal{E})$ and denote by $p$ the structure morphism $P\to Y$. Since by definition $\mathcal{E}=({\mathbb{S}}_{{\mathcal{O}}_Y}(\mathcal{E}){)}_1$, we have a canonical homomorphism ${\alpha }_1:\mathcal{E}\to p_{\ast }({\mathcal{O}}_P(1))$ (3.3.2.2), and therefore also (0, 4.4.3) a canonical homomorphism

$${\alpha }_1♯:p\ast (\mathcal{E})\to {\mathcal{O}}_P(1).(\mathrm{4.1.5.1})$$

Proposition.

The canonical homomorphism (4.1.5.1) is surjective.

Proof. We saw in (3.3.2) that ${\alpha }_1♯$ corresponds functorially to the canonical homomorphism $\mathcal{E}{\otimes }_{{\mathcal{O}}_Y}{\mathbb{S}}_{{\mathcal{O}}_Y}(\mathcal{E})\to ({\mathbb{S}}_{{\mathcal{O}}_Y}(\mathcal{E}))(1)$; since by definition $\mathcal{E}$ generates ${\mathbb{S}}_{{\mathcal{O}}_Y}(\mathcal{E})$, this homomorphism is surjective, whence the conclusion by (3.2.4).

4.2. Morphisms from a prescheme to a projective bundle

(4.2.1)

Keeping the notation of (4.1.5), let $X$ be a $Y$-prescheme, $q:X\to Y$ its structure morphism, and $r:X\to P$ a $Y$-morphism, so that we have the commutative diagram

         r
   P ←─────── X
    \       /
   p \     / q
      ↘   ↙
        Y

Since the functor $r\ast$ is right exact (0, 4.3.1), from the surjective homomorphism (4.1.5.1) we obtain a surjective homomorphism

$$r\ast ({\alpha }_1♯):r\ast (p\ast (\mathcal{E}))\to r\ast ({\mathcal{O}}_P(1)).$$

But $r\ast (p\ast (\mathcal{E}))=q\ast (\mathcal{E})$, and $r\ast ({\mathcal{O}}_P(1))$ is locally isomorphic to $r\ast ({\mathcal{O}}_P)={\mathcal{O}}_X$, in other words an invertible sheaf ${\mathcal{L}}_r$ on ${\mathcal{O}}_X$. We have thus defined, starting from $r$, a canonical surjective ${\mathcal{O}}_X$-homomorphism

$${\varphi }_r:q\ast (\mathcal{E})\to {\mathcal{L}}_r.(\mathrm{4.2.1.1})$$

When $Y=\mathrm{Spec}(A)$ is affine and $\mathcal{E}=\tilde{E}$, this homomorphism may be made more explicit as follows: given $f\in E$, it follows from (2.6.3) that

$$r^{-1}(D_{+}(f))=X_{{\varphi }_r♭(f)}.(\mathrm{4.2.1.2})$$

Let $V$ be an affine open of $X$ contained in $r^{-1}(D_{+}(f))$, and let $B$ be its ring, an $A$-algebra; set $S={\mathbb{S}}_A(E)$. The restriction of $r$ to $V$ corresponds to an $A$-homomorphism $\omega :S_{(f)}\to B$; we have $q\ast (\mathcal{E})|V=(E{\otimes }_{A}B)$ and ${\mathcal{L}}_r|V={\tilde{L}}_r$, where $L_r=(S(1){)}_{(f)}{\otimes }_{S_{(f)}}{B}_{[\omega ]}$ (I, 1.6.5). The restriction of ${\varphi }_r$ to $q\ast (\mathcal{E})|V$ corresponds to the $B$-homomorphism $u:E{\otimes }_{A}B\to L_r$ sending $x\otimes 1$ to $(x/1)\otimes f=(f/1)\otimes \omega (x/f)$. The canonical extension of ${\varphi }_r$ to a homomorphism of ${\mathcal{O}}_X$-algebras

  ψ_r : q*(𝕊(𝓔)) = 𝕊(q*(𝓔)) → 𝕊(ℒ_r) = ⊕_{n≥0} ℒ_r^{⊗n}

is thus such that the restriction of ${\psi }_r$ to $q\ast ({\mathbb{S}}_n(\mathcal{E}))|V$ corresponds to the homomorphism ${\mathbb{S}}_n(E{\otimes }_{A}B)={\mathbb{S}}_n(E){\otimes }_{A}B\to L_r^{\otimes n}$ sending $s\otimes 1$ to $(f/1{)}^{\otimes n}\otimes \omega (s/f^n)$.

(4.2.2)

Conversely, suppose given a morphism $q:X\to Y$, an invertible ${\mathcal{O}}_X$-module $\mathcal{L}$, and a quasi-coherent ${\mathcal{O}}_Y$-module $\mathcal{E}$; to a homomorphism $\varphi :q\ast (\mathcal{E})\to \mathcal{L}$ there canonically corresponds a homomorphism of quasi-coherent ${\mathcal{O}}_X$-algebras

  ψ : 𝕊(q*(𝓔)) = q*(𝕊(𝓔)) → ⊕_{n≥0} ℒ^{⊗n}

and therefore (3.7.1) a $Y$-morphism $r_{\mathcal{L},\psi }:G(\psi )\to \mathrm{Proj}(\mathbb{S}(\mathcal{E}))=\mathbb{P}(\mathcal{E})$, which we denote $r_{\mathcal{L},\varphi }$. If $\varphi$ is surjective, then so is $\psi$, and therefore (3.7.4) $r_{\mathcal{L},\varphi }$ is everywhere defined. Moreover, with the notation of (4.2.1) and (4.2.2):

Proposition.

Given a morphism $q:X\to Y$ and a quasi-coherent ${\mathcal{O}}_Y$-module $\mathcal{E}$, the maps $r\mapsto ({\mathcal{L}}_r,{\varphi }_r)$ and $(\mathcal{L},\varphi )\mapsto r_{\mathcal{L},\varphi }$ put into bijective correspondence the set of $Y$-morphisms $r:X\to \mathbb{P}(\mathcal{E})$ and the set of equivalence classes of pairs $(\mathcal{L},\varphi )$ consisting of an invertible ${\mathcal{O}}_X$-module $\mathcal{L}$ and a surjective homomorphism $\varphi :q\ast (\mathcal{E})\to \mathcal{L}$, two pairs $(\mathcal{L},\varphi )$ and $({\mathcal{L}}^{\prime },{\varphi }^{\prime })$ being equivalent if there exists an ${\mathcal{O}}_X$-isomorphism $\tau :\mathcal{L}\stackrel{\sim }{\to }{\mathcal{L}}^{\prime }$ such that ${\varphi }^{\prime }=\tau \circ \varphi$.

Proof. Start first from a $Y$-morphism $r:X\to \mathbb{P}(\mathcal{E})$, form ${\mathcal{L}}_r$ and ${\varphi }_r$ (4.2.1), and set $r^{\prime }=r_{{\mathcal{L}}_r,{\varphi }_r}$; it follows at once from (4.2.1) and (3.7.2) that the morphisms $r$ and $r^{\prime }$ are identical (taking as generator of ${\mathcal{L}}_r$ the element $(f/1)\otimes 1$ in order to define the homomorphisms $v_n$ of (3.7.2)). Conversely, start from a pair $(\mathcal{L},\varphi )$ and form

$r^{\prime \prime }=r_{\mathcal{L},\varphi }$, then ${\mathcal{L}}_{r^{\prime \prime }}$ and ${\varphi }_{r^{\prime \prime }}$; we show there is a canonical isomorphism $\tau :{\mathcal{L}}_{r^{\prime \prime }}\stackrel{\sim }{\to }\mathcal{L}$ such that $\varphi =\tau \circ {\varphi }_{r^{\prime \prime }}$. To define it we may place ourselves in the case $Y=\mathrm{Spec}(A)$, $X=\mathrm{Spec}(B)$ affine, and (with the notation of (4.2.1) and (3.7.2)) assign to each element $(x/1)\otimes 1$ of $L_{r^{\prime \prime }}$ (with $x\in E$) the element $v_1(x)c$ of $L$. One verifies at once that $\tau$ does not depend on the chosen generator $c$ of $L$; since $v_1$ is surjective by hypothesis, to prove $\tau$ is an isomorphism it suffices to show that if $x/1=0$ in $(S(1){)}_{(f)}$, then $v_1(x)/1=0$ in $B_g$; but the first relation says that $f^{n}x=0$ in ${\mathbb{S}}_{n+1}(E)$ for some $n$, whence $v_{n+1}(f^{n}x)=g^n{v}_1(x)=0$ in $B$, whence the conclusion. Finally, it is immediate that for two equivalent pairs $(\mathcal{L},\varphi )$ and $({\mathcal{L}}^{\prime },{\varphi }^{\prime })$ we have $r_{\mathcal{L},\varphi }=r_{{\mathcal{L}}^{\prime },{\varphi }^{\prime }}$.

In particular, for $X=Y$:

Theorem.

The set of $Y$-sections of $\mathbb{P}(\mathcal{E})$ is in canonical bijective correspondence with the set of quasi-coherent sub-${\mathcal{O}}_Y$-modules $\mathcal{F}$ of $\mathcal{E}$ such that $\mathcal{E}/\mathcal{F}$ is invertible.

Note that this property corresponds to the classical definition of "projective space" as the set of hyperplanes of a vector space (the classical case corresponds to $Y=\mathrm{Spec}(K)$, $K$ a field, and $\mathcal{E}=\tilde{E}$, $E$ a finite-dimensional $K$-vector space; the $\mathcal{F}$ with the property stated in (4.2.4) then correspond to the hyperplanes of $E$, and we know on the other hand that the $Y$-sections of $\mathbb{P}(\mathcal{E})$ are then the $K$-rational points of $\mathbb{P}(\mathcal{E})$ (I, 3.4.5)).

Remark.

Since there is a canonical bijective correspondence between $Y$-morphisms from $X$ to $P$ and their graph morphisms, the $X$-sections of $P{\times }_{Y}X$ (I, 3.3.14), we see that conversely (4.2.3) can be deduced from (4.2.4). Denote by $Hy{p}_Y(X,\mathcal{E})$ the set of quasi-coherent sub-${\mathcal{O}}_X$-modules $\mathcal{F}$ of $\mathcal{E}{\otimes }_Y{\mathcal{O}}_X=q\ast (\mathcal{E})$ such that $q\ast (\mathcal{E})/\mathcal{F}$ is an invertible ${\mathcal{O}}_X$-module. If $g:X^{\prime }\to X$ is a $Y$-morphism, the right-exactness of $g\ast$ gives $g\ast (q\ast (\mathcal{E})/\mathcal{F})=g\ast (q\ast (\mathcal{E}))/g\ast (\mathcal{F})$, so the latter sheaf is invertible, and therefore $Hy{p}_Y(X,\mathcal{E})$ is a contravariant functor on the category of $Y$-preschemes. The theorem (4.2.4) may then be interpreted as defining a canonical isomorphism of functors ${\mathrm{Hom}}_Y(X,\mathbb{P}(\mathcal{E}))$ and $Hy{p}_Y(X,\mathcal{E})$, contravariant in the variable $X$ over the category of $Y$-preschemes. This also gives a characterization of the projective bundle $P=\mathbb{P}(\mathcal{E})$ by the following universal property, closer to geometric intuition than the constructions of §§2 and 3: for every morphism $q:X\to Y$ and every invertible ${\mathcal{O}}_X$-module $\mathcal{L}$ that is a quotient of $\mathcal{E}{\otimes }_{{\mathcal{O}}_Y}{\mathcal{O}}_X$, there exists a unique $Y$-morphism $r:X\to P$ such that $\mathcal{L}=r\ast ({\mathcal{O}}_P(1))$.

We shall see later how, in the same way, one may define, among other things, the "Grassmannian" schemes.

Corollary.

Suppose every invertible ${\mathcal{O}}_Y$-module is trivial (I, 2.4.8). Let $V$ be the group ${\mathrm{Hom}}_{{\mathcal{O}}_Y}(\mathcal{E},{\mathcal{O}}_Y)$, regarded as a module over the ring $A=\Gamma (Y,{\mathcal{O}}_Y)$, and let $V\ast$ be the subset of $V$ formed by the surjective homomorphisms. Then the set of $Y$-sections of $\mathbb{P}(\mathcal{E})$ is canonically identified with $V\ast /A\ast$, where $A\ast$ is the group of units of $A$.

In particular:

1. Corollary (4.2.6) applies whenever $Y$ is a local scheme (I, 2.4.8). Let $Y$ be an arbitrary prescheme, $y$ a point of $Y$, and $Y^{\prime }=\mathrm{Spec}(\kappa (y))$; the fibre $p^{-1}(y)$ of $\mathbb{P}(\mathcal{E})$ is, by (4.1.3.1), identified with $\mathbb{P}({\mathcal{E}}^y)$, where ${\mathcal{E}}^y={\mathcal{E}}_y{\otimes }_{{\mathcal{O}}_y}\kappa (y)={\mathcal{E}}_y/{\mathfrak{m}}_y{\mathcal{E}}_y$ is regarded as a vector space over $\kappa (y)$. More generally, if $K$ is an extension of $\kappa (y)$, then $p^{-1}(y){\otimes }_{\kappa (y)}K$ is identified with $\mathbb{P}({\mathcal{E}}^y{\otimes }_{\kappa (y)}K)$. Corollary (4.2.6) therefore shows that the set of geometric points of $\mathbb{P}(\mathcal{E})$ with values in the extension $K$ of $\kappa (y)$ (I, 3.4.5), which one may also call the rational geometric fibre over $K$ of $\mathbb{P}(\mathcal{E})$ over the point $y$, is identified with the projective space associated to the dual of the $K$-vector space ${\mathcal{E}}^y{\otimes }_{\kappa (y)}K$.
2. Suppose $Y$ is affine of ring $A$, and that every invertible ${\mathcal{O}}_Y$-module is trivial; take in addition $\mathcal{E}={\mathcal{O}}_Y^n$. Then in (4.2.6), $V$ is identified with $A^n$ (I, 1.3.8), and $V\ast$ with the set of systems $(f_i{)}_{1\le i\le n}$ of elements of $A$ generating the ideal $A$; two such systems define the same $Y$-section of ${\mathbb{P}}_Y^{n-1}={\mathbb{P}}_A^{n-1}$ — in other words, the same point of ${\mathbb{P}}_A^{n-1}$ with values in $A$ — if and only if one is obtained from the other by multiplication by an invertible element of $A$.

These properties justify the terminology "projective bundle" for $\mathbb{P}(\mathcal{E})$. Note that the definition of "projective space" so obtained is in fact dual to the classical definition; this is forced upon us by the need to define $\mathbb{P}(\mathcal{E})$ for an arbitrary quasi-coherent ${\mathcal{O}}_Y$-module $\mathcal{E}$, not necessarily locally free.

Remark.

We shall see in Chapter V that, if $Y$ is locally Noetherian and connected and $\mathcal{E}$ is locally free, then, setting $P=\mathbb{P}(\mathcal{E})$, every invertible ${\mathcal{O}}_P$-module $\mathcal{L}$ is isomorphic to one of the form ${\mathcal{L}}^{\prime }{\otimes }_{{\mathcal{O}}_Y}{\mathcal{O}}_P(m)$, where ${\mathcal{L}}^{\prime }$ is an invertible ${\mathcal{O}}_Y$-module, well-determined up to isomorphism, and $m$ is a well-determined integer. In other words, $H^1(P,{\mathcal{O}}_P\ast )$ is isomorphic to $\mathbb{Z}\times H^1(Y,{\mathcal{O}}_Y\ast )$ (0, 5.4.7). We shall also see (III, 2.1.14, taking (0, 5.4.10) into account) that $p_{\ast }({\mathcal{L}}^{\otimes m})=0$ for $m<0$ and $p_{\ast }({\mathcal{L}}^{\otimes m})$ is isomorphic to ${\mathcal{L}}^{\prime }{\otimes }_{{\mathcal{O}}_Y}({\mathbb{S}}_{{\mathcal{O}}_Y}(\mathcal{E}){)}_m$ for $m\ge 0$. If $\mathcal{F}$ is a quasi-coherent ${\mathcal{O}}_Y$-module, every $Y$-morphism $\mathbb{P}(\mathcal{F})\to \mathbb{P}(\mathcal{E})$ is therefore determined by the data of an invertible ${\mathcal{O}}_Y$-module ${\mathcal{L}}^{\prime }$, an integer $m\ge 0$, and an ${\mathcal{O}}_Y$-homomorphism $\psi :\mathcal{F}\to {\mathcal{L}}^{\prime }{\otimes }_{{\mathcal{O}}_Y}({\mathbb{S}}_{{\mathcal{O}}_Y}(\mathcal{E}){)}_m$ such that the corresponding homomorphism $\psi ♯$ of ${\mathcal{O}}_{\mathbb{P}(\mathcal{F})}$-modules is surjective. We shall also see that if the $Y$-morphism in question is an isomorphism, then $m=1$ and $\mathcal{F}$ is isomorphic to $\mathcal{E}{\otimes }_{{\mathcal{O}}_Y}{\mathcal{L}}^{\prime }$ (the converse of (4.1.4)). This will let us determine the sheaf of germs of automorphisms of $\mathbb{P}(\mathcal{E})$ as the quotient of the sheaf of groups $\mathrm{Aut}(\mathcal{E})$ (which is locally isomorphic to $GL(n,{\mathcal{O}}_Y)$ if $\mathcal{E}$ is of rank $n$) by ${\mathcal{O}}_Y\ast$.

(4.2.8)

Keeping the notation of (4.2.1), let $u:X^{\prime }\to X$ be a morphism; if the $Y$-morphism $r:X\to P$ corresponds to the homomorphism $\varphi :q\ast (\mathcal{E})\to \mathcal{L}$, then the $Y$-morphism $r\circ u$ corresponds to $u\ast (\varphi ):u\ast (q\ast (\mathcal{E}))\to u\ast (\mathcal{L})$, as follows immediately from the definitions.

(4.2.9)

Let $\mathcal{E}$, $\mathcal{F}$ be two quasi-coherent ${\mathcal{O}}_Y$-modules, $v:\mathcal{E}\to \mathcal{F}$ a surjective homomorphism, and $j=\mathbb{P}(v)$ the corresponding closed immersion $\mathbb{P}(\mathcal{F})\to \mathbb{P}(\mathcal{E})$ (4.1.2). If the $Y$-morphism $r:X\to \mathbb{P}(\mathcal{F})$ corresponds to the homomorphism $\varphi :q\ast (\mathcal{F})\to \mathcal{L}$, then the

$Y$-morphism $j\circ r$ corresponds to $q\ast (\mathcal{E})\stackrel{q\ast (v)}{\to }q\ast (\mathcal{F})\stackrel{\varphi }{\to }\mathcal{L}$; this again follows from the definition given in (4.2.1).

(4.2.10)

Let $\psi :Y^{\prime }\to Y$ be a morphism, and set ${\mathcal{E}}^{\prime }=\psi \ast (\mathcal{E})$. If the $Y$-morphism $r:X\to P$ corresponds to the homomorphism $\varphi :q\ast (\mathcal{E})\to \mathcal{L}$, then the $Y^{\prime }$-morphism

  r_{(Y')} : X_{(Y')} → P' = ℙ(𝓔')

corresponds to ${\varphi }_{(Y^{\prime })}:q_{(Y^{\prime })}\ast ({\mathcal{E}}^{\prime })=q\ast (\mathcal{E}){\otimes }_Y{\mathcal{O}}_{Y^{\prime }}\to \mathcal{L}{\otimes }_Y{\mathcal{O}}_{Y^{\prime }}$. Indeed, by (4.1.3.1) we have the commutative diagram

   Y' ←─── P' = P_{(Y')} ←─── X_{(Y')}
   │           │                │
   │           │ u              │ v
   ↓           ↓                ↓
   Y  ←─── P            ←─── X
              p              r

By (4.1.3.2),

  (r_{(Y')})*(𝒪_{P'}(1)) = (r_{(Y')})*(u*(𝒪_P(1))) = v*(r*(𝒪_P(1)))
                         = v*(ℒ) = ℒ ⊗_Y 𝒪_{Y'};

on the other hand, $u\ast ({\alpha }_1♯)$ is precisely the canonical homomorphism ${\alpha }_1♯:(p_{(Y^{\prime })})\ast ({\mathcal{E}}^{\prime })\to {\mathcal{O}}_{P^{\prime }}(1)$, as one sees by making the canonical homomorphisms ${\alpha }_1♯$ for $P$ and $P^{\prime }$ explicit as in (4.1.6). Whence our assertion.

4.3. The Segre morphism

(4.3.1)

Let $Y$ be a prescheme and $\mathcal{E}$, $\mathcal{F}$ two quasi-coherent ${\mathcal{O}}_Y$-modules; set $P_1=\mathbb{P}(\mathcal{E})$, $P_2=\mathbb{P}(\mathcal{F})$, and let $p_1:P_1\to Y$, $p_2:P_2\to Y$ be the structure morphisms. Let $Q=P_1{\times }_Y{P}_2$, and let $q_1:Q\to P_1$, $q_2:Q\to P_2$ be the canonical projections; the ${\mathcal{O}}_Q$-module $\mathcal{L}={\mathcal{O}}_{P_1}(1){\otimes }_Y{\mathcal{O}}_{P_2}(1)=q_1\ast ({\mathcal{O}}_{P_1}(1)){\otimes }_{{\mathcal{O}}_Q}{q}_2\ast ({\mathcal{O}}_{P_2}(1))$ is invertible, as the tensor product of two invertible ${\mathcal{O}}_Q$-modules (0, 5.4.4). On the other hand, if $r=p_1\circ q_1=p_2\circ q_2$ is the structure morphism $Q\to Y$, then $r\ast (\mathcal{E}{\otimes }_{{\mathcal{O}}_Y}\mathcal{F})=q_1\ast (p_1\ast (\mathcal{E})){\otimes }_{{\mathcal{O}}_Q}{q}_2\ast (p_2\ast (\mathcal{F}))$ (0, 4.3.3). The canonical surjective homomorphisms (4.1.5.1) $p_1\ast (\mathcal{E})\to {\mathcal{O}}_{P_1}(1)$ and $p_2\ast (\mathcal{F})\to {\mathcal{O}}_{P_2}(1)$ therefore yield, by tensor product, a canonical homomorphism

  s : r*(𝓔 ⊗_{𝒪_Y} 𝓕) → ℒ                                                   (4.3.1.1)

which is evidently surjective; from this we obtain (4.2.2) a canonical morphism, called the Segre morphism:

  ς : ℙ(𝓔) ×_Y ℙ(𝓕) → ℙ(𝓔 ⊗_{𝒪_Y} 𝓕).                                       (4.3.1.2)

Let us make $\varsigma$ explicit when $Y=\mathrm{Spec}(A)$ is affine, $\mathcal{E}=E$, $\mathcal{F}=F$, with $E$, $F$ two $A$-modules, so that $\mathcal{E}{\otimes }_{{\mathcal{O}}_Y}\mathcal{F}=(E{\otimes }_{A}F)$ (I, 1.3.12); set $R={\mathbb{S}}_A(E)$, $S={\mathbb{S}}_A(F)$, $T={\mathbb{S}}_A(E{\otimes }_{A}F)$. Let $f\in E$, $g\in F$, and consider the affine open

  D₊(f) ×_Y D₊(g) = Spec(B)

of $Q$, where $B=R_{(f)}{\otimes }_A{S}_{(g)}$; the restriction of $\mathcal{L}$ to this affine open is $\tilde{L}$, where

  L = (R(1))_{(f)} ⊗_A (S(1))_{(g)},

and the element $c=(f/1)\otimes (g/1)$ is a generator of $L$, regarded as a free $B$-module (2.5.7). The homomorphism (4.3.1.1) corresponds to the homomorphism

  (x ⊗ y) ⊗ b ↦ b ((x/1) ⊗ (y/1))

from $(E{\otimes }_{A}F){\otimes }_{A}B$ to $L$. With the notation of (3.7.2) we therefore have $v_1(x\otimes y)=(x/f)\otimes (y/g)$; the restriction of $\varsigma$ to $D_{+}(f){\times }_Y{D}_{+}(g)$ is a morphism of this affine scheme to $D_{+}(f\otimes g)$, corresponding to the ring homomorphism $\omega :T_{(f\otimes g)}\to R_{(f)}{\otimes }_A{S}_{(g)}$ defined by

  ω((x ⊗ y)/(f ⊗ g)) = (x/f) ⊗ (y/g)                                       (4.3.1.3)

for $x\in E$ and $y\in F$.

(4.3.2)

It follows from (4.2.3) that we have a canonical isomorphism

  τ : ς*(𝒪_P(1)) ⥲ 𝒪_{P_1}(1) ⊗_Y 𝒪_{P_2}(1)                              (4.3.2.1)

where we have set $P=\mathbb{P}(\mathcal{E}{\otimes }_{{\mathcal{O}}_Y}\mathcal{F})$. We show that, for $x\in \Gamma (Y,\mathcal{E})$ and $y\in \Gamma (Y,\mathcal{F})$,

  τ(α_1(x ⊗ y)) = α_1(x) ⊗ α_1(y).                                         (4.3.2.2)

Indeed, we reduce to the case $Y$ affine, and we then have, with the notation of (4.3.1) and (2.6.2), ${\alpha }_1^{f\otimes g}(x\otimes y)=(x\otimes y)/1$ in $(T(1){)}_{(f\otimes g)}$, ${\alpha }_1^f(x)=x/1$ in $(R(1){)}_{(f)}$, and ${\alpha }_1^g(y)=y/1$ in $(S(1){)}_{(g)}$. The definition of $\tau$ given in (4.2.3) and the computation of $v_1$ done in (4.3.1) prove (4.3.2.2) at once. From this we derive

  ς⁻¹(P_{x⊗y}) = (P_1)_x ×_Y (P_2)_y                                        (4.3.2.3)

with the notation of (3.1.4). Indeed, taking (3.3.3) into account, the formula (4.3.2.2) reduces (returning to the affine case via (I, 3.2.7) and (I, 3.2.3)) to proving the following lemma:

Lemma.

Let $B$, $B^{\prime }$ be two $A$-algebras, and set $Y=\mathrm{Spec}(A)$, $Z=\mathrm{Spec}(B)$, $Z^{\prime }=\mathrm{Spec}(B^{\prime })$. For any $t\in B$ and $t^{\prime }\in B^{\prime }$, we have $D(t\otimes t^{\prime })=D(t){\times }_{Y}D(t^{\prime })$.

Proof. If $p:Z{\times }_Y{Z}^{\prime }\to Z$ and $p^{\prime }:Z{\times }_Y{Z}^{\prime }\to Z^{\prime }$ are the canonical projections, it follows from (I, 1.2.2.2) that $p^{-1}(D(t))=D(t\otimes 1)$ and $p^{\prime -1}(D(t^{\prime }))=D(1\otimes t^{\prime })$; the conclusion follows from (I, 3.2.7) and (I, 1.1.9.1), since $(t\otimes 1)(1\otimes t^{\prime })=t\otimes t^{\prime }$.

Proposition.

The Segre morphism is a closed immersion.

Proof. The question being local on $Y$, we reduce to the case where $Y$ is affine. With the notation of (4.3.1) and (4.3.2), the $D_{+}(f\otimes g)$ form a basis for the topology of $P$, since the $f\otimes g$ generate $T$ when $f$ ranges over $E$ and $g$ over $F$. On the other hand, by (4.3.2.3), ${\varsigma }^{-1}(D_{+}(f\otimes g))=D_{+}(f){\times }_Y{D}_{+}(g)$. It thus suffices (I, 4.2.4) to prove that the restriction of $\varsigma$ to $D_{+}(f){\times }_Y{D}_{+}(g)$ is a closed immersion into $D_{+}(f\otimes g)$. But, with the same notation, formula (4.3.1.3) shows that $\omega$ is surjective, which completes the proof.

(4.3.4)

The Segre morphism is functorial in $\mathcal{E}$ and $\mathcal{F}$, when one restricts the

homomorphisms of quasi-coherent ${\mathcal{O}}_Y$-modules to surjective homomorphisms. Indeed, we must show that if $\mathcal{E}\to {\mathcal{E}}^{\prime }$ is a surjective ${\mathcal{O}}_Y$-homomorphism, then the diagram

                       j × 1
   ℙ(𝓔') × ℙ(𝓕) ─────────────→ ℙ(𝓔) × ℙ(𝓕)

         │                          │
       ς │                          │ ς
         ↓                          ↓

   ℙ(𝓔' ⊗ 𝓕) ────────────────→ ℙ(𝓔 ⊗ 𝓕)

commutes, $j$ denoting the canonical closed immersion $\mathbb{P}({\mathcal{E}}^{\prime })\to \mathbb{P}(\mathcal{E})$. Set $P_1^{\prime }=\mathbb{P}({\mathcal{E}}^{\prime })$ and keep the other notation of (4.3.1); $j\times 1$ is a closed immersion (I, 4.3.1), and up to isomorphism

$$(j\times 1)\ast ({\mathcal{O}}_{P_1}(1)\otimes {\mathcal{O}}_{P_2}(1))=j\ast ({\mathcal{O}}_{P_1}(1))\otimes {\mathcal{O}}_{P_2}(1)={\mathcal{O}}_{P_1^{\prime }}(1)\otimes {\mathcal{O}}_{P_2}(1)$$

by (4.1.2.1) and (I, 9.1.5); our assertion therefore follows from (4.2.8) and (4.2.9).

(4.3.5)

With the notation of (4.3.1), let $\psi :Y^{\prime }\to Y$ be a morphism, and set ${\mathcal{E}}^{\prime }=\psi \ast (\mathcal{E})$, ${\mathcal{F}}^{\prime }=\psi \ast (\mathcal{F})$. Then the Segre morphism $\mathbb{P}({\mathcal{E}}^{\prime })\times \mathbb{P}({\mathcal{F}}^{\prime })\to \mathbb{P}({\mathcal{E}}^{\prime }\otimes {\mathcal{F}}^{\prime })$ is identified with ${\varsigma }_{(Y^{\prime })}$. Indeed, keeping the notation of (4.3.1), set in addition $P_1^{\prime }=\mathbb{P}({\mathcal{E}}^{\prime })$, $P_2^{\prime }=\mathbb{P}({\mathcal{F}}^{\prime })$; we know (4.1.3.1) that $P_i^{\prime }$ is identified with $(P_i{)}_{(Y^{\prime })}$ ($i=1,2$), so the structure morphism $P_1^{\prime }\times P_2^{\prime }\to Y^{\prime }$ is identified with $r_{(Y^{\prime })}$. On the other hand, ${\mathcal{E}}^{\prime }\otimes {\mathcal{F}}^{\prime }$ is identified with $\psi \ast (\mathcal{E}\otimes \mathcal{F})$, so $\mathbb{P}({\mathcal{E}}^{\prime }\otimes {\mathcal{F}}^{\prime })$ is identified with $(\mathbb{P}(\mathcal{E}\otimes \mathcal{F}){)}_{(Y^{\prime })}$ (4.1.3.1). Finally, ${\mathcal{O}}_{P_1^{\prime }}(1){\otimes }_{Y^{\prime }}{\mathcal{O}}_{P_2^{\prime }}(1)={\mathcal{L}}^{\prime }$ is identified with $\mathcal{L}{\otimes }_Y{\mathcal{O}}_{Y^{\prime }}$, by (4.1.3.2) and (I, 9.1.11). The canonical homomorphism $(r_{(Y^{\prime })})\ast ({\mathcal{E}}^{\prime }\otimes {\mathcal{F}}^{\prime })\to {\mathcal{L}}^{\prime }$ is then identified with $s_{(Y^{\prime })}$, and our assertion follows from (4.2.10).

Remark.

The prescheme given by the disjoint sum of $\mathbb{P}(\mathcal{E})$ and $\mathbb{P}(\mathcal{F})$ is likewise canonically isomorphic to a closed subprescheme of $\mathbb{P}(\mathcal{E}\oplus \mathcal{F})$. Indeed, the surjective homomorphisms $\mathcal{E}\oplus \mathcal{F}\to \mathcal{E}$ and $\mathcal{E}\oplus \mathcal{F}\to \mathcal{F}$ correspond to closed immersions $\mathbb{P}(\mathcal{E})\to \mathbb{P}(\mathcal{E}\oplus \mathcal{F})$ and $\mathbb{P}(\mathcal{F})\to \mathbb{P}(\mathcal{E}\oplus \mathcal{F})$; everything comes down to showing that the underlying spaces of the closed subpreschemes of $\mathbb{P}(\mathcal{E}\oplus \mathcal{F})$ so obtained have no point in common. The question being local on $Y$, we may adopt the notation of (4.3.1); now ${\mathbb{S}}_n(E)$ and ${\mathbb{S}}_n(F)$ are identified with submodules of ${\mathbb{S}}_n(E\oplus F)$ whose intersection reduces to 0; and if $\mathfrak{p}$ is a graded prime ideal of $\mathbb{S}(E)$ such that $\mathfrak{p}\cap {\mathbb{S}}_n(E)\ne {\mathbb{S}}_n(E)$ for every $n\ge 0$, then it corresponds in $\mathbb{S}(E\oplus F)$ to a graded prime ideal whose trace on ${\mathbb{S}}_n(E)$ is $\mathfrak{p}\cap {\mathbb{S}}_n(E)$, but which contains ${\mathbb{S}}_{+}(F)$, as one sees at once. Hence two points of $\mathrm{Proj}(\mathbb{S}(E))$ and $\mathrm{Proj}(\mathbb{S}(F))$ respectively cannot have the same image in $\mathrm{Proj}(\mathbb{S}(E\oplus F))$.

4.4. Immersions into projective bundles. Very ample sheaves

Proposition.

Let $Y$ be a quasi-compact scheme or a prescheme whose underlying space is Noetherian, $q:X\to Y$ a morphism of finite type, and $\mathcal{L}$ an invertible ${\mathcal{O}}_X$-module.

1. Let $\mathcal{S}$ be a quasi-coherent graded ${\mathcal{O}}_Y$-algebra with positive degrees, and $\psi :q\ast (\mathcal{S})\to {\oplus }_{n\ge 0}{\mathcal{L}}^{\otimes n}$ a homomorphism of graded algebras. For $r_{\mathcal{L},\psi }$ to be everywhere defined and an immersion, it is necessary

and sufficient that there exist an integer `n ≥ 0` and a quasi-coherent
sub-`𝒪_Y`-module `𝓔` of `𝒮_n` _of finite type_ such that the
homomorphism
`φ' = ψ_n ∘ q*(j) : q*(𝓔) → ℒ^{⊗n} = ℒ'`
(with `j` the injection `𝓔 → 𝒮_n`) is surjective and the morphism
`r_{ℒ',φ'} : X → ℙ(𝓔)` is an immersion.

1. Let $\mathcal{F}$ be a quasi-coherent ${\mathcal{O}}_Y$-module, and $\varphi :q\ast (\mathcal{F})\to \mathcal{L}$ a surjective homomorphism. For the morphism $r_{\mathcal{L},\varphi }$ to be an immersion $X\to \mathbb{P}(\mathcal{F})$, it is necessary and sufficient that there exist a quasi-coherent sub-${\mathcal{O}}_Y$-module $\mathcal{E}$ of $\mathcal{F}$ of finite type such that the homomorphism ${\varphi }^{\prime }=\varphi \circ q\ast (j):q\ast (\mathcal{E})\to \mathcal{L}$ (with $j:\mathcal{E}\to \mathcal{F}$ the canonical injection) is surjective and $r_{\mathcal{L},{\varphi }^{\prime }}:X\to \mathbb{P}(\mathcal{E})$ is an immersion.

Proof.

(i) The fact that $r_{\mathcal{L},\psi }$ is everywhere defined and an immersion is equivalent, by (3.8.5), to the existence of an $n\ge 0$ and an $\mathcal{E}$ such that, if ${\mathcal{S}}^{\prime }$ is the sub-algebra of $\mathcal{S}$ generated by $\mathcal{E}$, the homomorphism $q\ast (\mathcal{E})\to {\mathcal{L}}^{\otimes n}$ is surjective and $r_{\mathcal{L},{\psi }^{\prime }}:X\to \mathrm{Proj}({\mathcal{S}}^{\prime })$ is everywhere defined and an immersion. We have on the other hand a canonical surjective homomorphism $\mathbb{S}(\mathcal{E})\to {\mathcal{S}}^{\prime }$, to which corresponds a closed immersion $\mathrm{Proj}({\mathcal{S}}^{\prime })\to \mathbb{P}(\mathcal{E})$ (3.6.2); whence the conclusion.

(ii) Since $\mathcal{F}$ is the direct limit of its quasi-coherent submodules of finite type ${\mathcal{E}}_{\lambda }$ (I, 9.4.9), $\mathbb{S}(\mathcal{F})$ is the direct limit of the $\mathbb{S}({\mathcal{E}}_{\lambda })$; the conclusion follows from (3.8.4), upon observing that in the proof of (3.8.4) one may take all the $n_i$ equal to 1: indeed (assuming $Y$ affine), if $r=r_{\mathcal{L},\varphi }$ is an immersion, then $r(X)$ is a quasi-compact subspace of $\mathbb{P}(\mathcal{F})$ that may be covered by finitely many open subsets of $\mathbb{P}(\mathcal{F})$ of the form $D_{+}(f)$ with $f\in F$, such that $D_{+}(f)\cap r(X)$ is closed.

Definition.

Let $Y$ be a prescheme, $q:X\to Y$ a morphism. We say that an invertible ${\mathcal{O}}_X$-module $\mathcal{L}$ is very ample for $q$ (or very ample for $Y$, or simply very ample if no confusion results) if there exists a quasi-coherent ${\mathcal{O}}_Y$-module $\mathcal{E}$ and a $Y$-immersion $i$ of $X$ into $P=\mathbb{P}(\mathcal{E})$ such that $\mathcal{L}$ is isomorphic to $i\ast ({\mathcal{O}}_P(1))$.

It is equivalent (4.2.3) to say that there exists a quasi-coherent ${\mathcal{O}}_Y$-module $\mathcal{E}$ and a surjective homomorphism $\varphi :q\ast (\mathcal{E})\to \mathcal{L}$ such that $r_{\mathcal{L},\varphi }:X\to \mathbb{P}(\mathcal{E})$ is an immersion.

Note that the existence of a very-ample-for-$Y$ ${\mathcal{O}}_X$-module implies that $q$ is separated ((3.1.3) and (I, 5.5.1, (i) and (ii))).

Corollary.

Suppose there exists a quasi-coherent graded ${\mathcal{O}}_Y$-algebra $\mathcal{S}$ generated by ${\mathcal{S}}_1$ and a $Y$-immersion $i:X\to P=\mathrm{Proj}(\mathcal{S})$ such that $\mathcal{L}$ is isomorphic to $i\ast ({\mathcal{O}}_P(1))$; then $\mathcal{L}$ is very ample relative to $q$.

Proof. If $\mathcal{F}={\mathcal{S}}_1$, the canonical homomorphism $\mathbb{S}(\mathcal{F})\to \mathcal{S}$ is surjective; composing the corresponding closed immersion $\mathrm{Proj}(\mathcal{S})\to \mathbb{P}(\mathcal{F})$ (3.6.2) with the immersion $i$, we obtain an immersion $j:X\to \mathbb{P}(\mathcal{F})=P^{\prime }$ such that $\mathcal{L}$ is isomorphic to $j\ast ({\mathcal{O}}_{P^{\prime }}(1))$ (3.6.3).

Proposition.

Let $q:X\to Y$ be a quasi-compact morphism, and $\mathcal{L}$ an invertible ${\mathcal{O}}_X$-module. The following properties are equivalent:

a) $\mathcal{L}$ is very ample relative to $q$. b) $q_{\ast }(\mathcal{L})$ is quasi-coherent, the canonical homomorphism $\sigma :q\ast (q_{\ast }(\mathcal{L}))\to \mathcal{L}$ is surjective, and the morphism $r_{\mathcal{L},\sigma }:X\to \mathbb{P}(q_{\ast }(\mathcal{L}))$ is an immersion.

Proof. Since $q$ is quasi-compact, we know $q_{\ast }(\mathcal{L})$ is quasi-coherent when $q$ is separated (I, 9.2.2).

We know (3.4.7) that the existence of a surjective homomorphism $\varphi :q\ast (\mathcal{E})\to \mathcal{L}$ (with $\mathcal{E}$ a quasi-coherent ${\mathcal{O}}_Y$-module) implies that $\sigma$ is surjective; moreover, to the factorization $q\ast (\mathcal{E})\to q\ast (q_{\ast }(\mathcal{L}))\stackrel{\sigma }{\to }\mathcal{L}$ of $\varphi$ corresponds canonically a factorization

  q*(𝕊(𝓔)) → q*(𝕊(q_*(ℒ))) → ⊕_{n≥0} ℒ^{⊗n}

so (3.8.3) the hypothesis that $r_{\mathcal{L},\varphi }$ is an immersion implies that so is $j=r_{\mathcal{L},\sigma }$; moreover (4.2.4), $\mathcal{L}$ is isomorphic to $j\ast ({\mathcal{O}}_{P^{\prime }}(1))$ with $P^{\prime }=\mathbb{P}(q_{\ast }(\mathcal{L}))$. Thus (a) and (b) are equivalent.

Corollary.

Suppose $q$ is quasi-compact. For $\mathcal{L}$ to be very ample relative to $Y$, it is necessary and sufficient that there exist an open cover $(U_{\alpha })$ of $Y$ such that $\mathcal{L}|q^{-1}(U_{\alpha })$ is very ample relative to $U_{\alpha }$ for every $\alpha$.

Proof. Indeed, condition (b) of (4.4.4) is local on $Y$.

Proposition.

Let $Y$ be a quasi-compact scheme or a prescheme whose underlying space is Noetherian, $q:X\to Y$ a morphism of finite type, and $\mathcal{L}$ an invertible ${\mathcal{O}}_X$-module. Then conditions (a) and (b) of (4.4.4) are equivalent to the following:

a') There exists a quasi-coherent ${\mathcal{O}}_Y$-module $\mathcal{E}$ of finite type and a surjective homomorphism $\varphi :q\ast (\mathcal{E})\to \mathcal{L}$ such that $r_{\mathcal{L},\varphi }$ is an immersion. b') There exists a coherent sub-${\mathcal{O}}_Y$-module $\mathcal{E}$ of $q_{\ast }(\mathcal{L})$ of finite type with the properties stated in (a').

Proof. It is clear that (a') or (b') implies (a); on the other hand, (a) implies (a') by (4.4.1), and similarly (b) implies (b').

Corollary.

Suppose $Y$ is a quasi-compact scheme or a Noetherian prescheme. If $\mathcal{L}$ is very ample for $q$, then there exists a quasi-coherent graded ${\mathcal{O}}_Y$-algebra $\mathcal{S}$ such that ${\mathcal{S}}_1$ is of finite type and generates $\mathcal{S}$, and a dominant open $Y$-immersion $i:X\to P=\mathrm{Proj}(\mathcal{S})$ such that $\mathcal{L}$ is isomorphic to $i\ast ({\mathcal{O}}_P(1))$.

Proof. Condition (b') of (4.4.6) is satisfied; the structure morphism $p:\mathbb{P}(\mathcal{E})=P^{\prime }\to Y$ is then separated and of finite type (3.1.3), so $P^{\prime }$ is a quasi-compact scheme (resp. a Noetherian prescheme) if $Y$ is one. Let $Z$ be the closure (I, 9.5.11) of the subprescheme $X^{\prime }$ of $P^{\prime }$ associated to the immersion $j=r_{\mathcal{L},\varphi }$ of $X$ into $P^{\prime }$; it is clear that $j$ factors as the canonical injection $Z\to P^{\prime }$ followed by a dominant open immersion $i:X\to Z$. But $Z$ is identified with a prescheme $\mathrm{Proj}(\mathcal{S})$, where $\mathcal{S}$ is a graded ${\mathcal{O}}_Y$-algebra quotient of $\mathbb{S}(\mathcal{E})$ by a quasi-coherent graded sheaf of ideals (3.6.2); it is clear that ${\mathcal{S}}_1$ is of finite type and generates $\mathcal{S}$. Moreover, ${\mathcal{O}}_Z(1)$ is the inverse image of ${\mathcal{O}}_{P^{\prime }}(1)$ under the canonical injection (3.6.3), so $\mathcal{L}=i\ast ({\mathcal{O}}_Z(1))$.

Proposition.

Let $q:X\to Y$ be a morphism, $\mathcal{L}$ a very-ample-relative-to-$q$ ${\mathcal{O}}_X$-module, and ${\mathcal{L}}^{\prime }$ an invertible ${\mathcal{O}}_X$-module such that there exist a quasi-coherent ${\mathcal{O}}_Y$-module ${\mathcal{E}}^{\prime }$ and a surjective homomorphism $q\ast ({\mathcal{E}}^{\prime })\to {\mathcal{L}}^{\prime }$. Then $\mathcal{L}{\otimes }_{{\mathcal{O}}_X}{\mathcal{L}}^{\prime }$ is very ample relative to $q$.

Proof. The hypothesis implies the existence of a $Y$-morphism $r^{\prime }:X\to P^{\prime }=\mathbb{P}({\mathcal{E}}^{\prime })$ such that ${\mathcal{L}}^{\prime }=r^{\prime }\ast ({\mathcal{O}}_{P^{\prime }}(1))$ (4.2.1). By hypothesis there is also a quasi-coherent ${\mathcal{O}}_Y$-module $\mathcal{E}$ and a

$Y$-immersion $r:X\to P=\mathbb{P}(\mathcal{E})$ such that $\mathcal{L}=r\ast ({\mathcal{O}}_P(1))$. Set $Q=\mathbb{P}(\mathcal{E}\otimes {\mathcal{E}}^{\prime })$ and consider the Segre morphism $\varsigma :P{\times }_Y{P}^{\prime }\to Q$ (4.3.1). Since $r$ is an immersion, so is $(r,r^{\prime }{)}_Y:X\to P{\times }_Y{P}^{\prime }$ (I, 5.3.14); but $\varsigma$ is an immersion (4.3.3), so $r^{\prime \prime }:X\stackrel{(r,r^{\prime })}{\to }P{\times }_Y{P}^{\prime }\stackrel{\varsigma }{\to }Q$ is too. On the other hand (4.3.2.1), $\varsigma \ast ({\mathcal{O}}_Q(1))$ is isomorphic to ${\mathcal{O}}_P(1){\otimes }_Y{\mathcal{O}}_{P^{\prime }}(1)$, so (I, 9.1.4) $r^{\prime \prime }\ast ({\mathcal{O}}_Q(1))$ is isomorphic to $\mathcal{L}\otimes {\mathcal{L}}^{\prime }$, which proves the proposition.

Corollary.

Let $q:X\to Y$ be a morphism.

1. Let $\mathcal{L}$ be an invertible ${\mathcal{O}}_X$-module and $\mathcal{K}$ an invertible ${\mathcal{O}}_Y$-module. For $\mathcal{L}$ to be very ample relative to $q$, it is necessary and sufficient that $\mathcal{L}\otimes q\ast (\mathcal{K})$ be so.
2. If $\mathcal{L}$ and ${\mathcal{L}}^{\prime }$ are two ${\mathcal{O}}_X$-modules very ample relative to $q$, then so is $\mathcal{L}\otimes {\mathcal{L}}^{\prime }$; in particular, ${\mathcal{L}}^{\otimes n}$ is very ample relative to $q$ for every $n>0$.

Proof. (ii) is an immediate consequence of (4.4.8), as is the necessity of (i); on the other hand, if $\mathcal{L}\otimes q\ast (\mathcal{K})$ is very ample, so is $(\mathcal{L}\otimes q\ast (\mathcal{K}))\otimes q\ast ({\mathcal{K}}^{-1})$ by the above, and this latter ${\mathcal{O}}_X$-module is isomorphic to $\mathcal{L}$ (0, 5.4.3 and 5.4.5).

Proposition.

1. For every prescheme $Y$, every invertible ${\mathcal{O}}_Y$-module $\mathcal{L}$ is very ample relative to the identity morphism 1_Y.
2. (i bis) Let $f:X\to Y$ be a morphism and $j:X^{\prime }\to X$ an immersion. If $\mathcal{L}$ is an ${\mathcal{O}}_X$-module very ample relative to $f$, then $j\ast (\mathcal{L})$ is very ample relative to $f\circ j$.
3. Let $Z$ be a quasi-compact prescheme, $f:X\to Y$ a morphism of finite type, $g:Y\to Z$ a quasi-compact morphism, $\mathcal{L}$ an ${\mathcal{O}}_X$-module very ample relative to $f$, and $\mathcal{K}$ an ${\mathcal{O}}_Y$-module very ample relative to $g$. Then there exists an integer $n_0>0$ such that $\mathcal{L}\otimes f\ast ({\mathcal{K}}^{\otimes n})$ is very ample relative to $g\circ f$ for all $n\ge n_0$.
4. Let $f:X\to Y$ and $g:Y^{\prime }\to Y$ be morphisms, and set $X^{\prime }=X_{(Y^{\prime })}$. If $\mathcal{L}$ is an ${\mathcal{O}}_X$-module very ample relative to $f$, then ${\mathcal{L}}^{\prime }=\mathcal{L}{\otimes }_Y{\mathcal{O}}_{Y^{\prime }}$ is an ${\mathcal{O}}_{X^{\prime }}$-module very ample relative to $f_{(Y^{\prime })}$.
5. Let $f_i:X_i\to Y_i$ ($i=1,2$) be two $S$-morphisms. If ${\mathcal{L}}_i$ is an ${\mathcal{O}}_{X_i}$-module very ample relative to $f_i$ ($i=1,2$), then ${\mathcal{L}}_1{\otimes }_S{\mathcal{L}}_2$ is very ample relative to $f_1{\times }_S{f}_2$.
6. Let $f:X\to Y$ and $g:Y\to Z$ be morphisms. If an ${\mathcal{O}}_X$-module $\mathcal{L}$ is very ample relative to $g\circ f$, then $\mathcal{L}$ is very ample relative to $f$.
7. Let $f:X\to Y$ be a morphism, and $j$ the canonical injection $X_{red}\to X$. If $\mathcal{L}$ is an ${\mathcal{O}}_X$-module very ample relative to $f$, then $j\ast (\mathcal{L})$ is very ample relative to $f_{red}$.

Proof. Property (i bis) follows immediately from Definition (4.4.2), and it is immediate that (vi) follows formally from (i bis) and (v) by an argument modeled on (I, 5.5.12), which we leave to the reader. To prove (v), consider, as in (I, 5.5.12), the factorization $X\stackrel{{\Gamma }_f}{\to }X{\times }_{Z}Y\stackrel{p_2}{\to }Y$, noting that $p_2=(g\circ f)\times 1_Y$. It follows from the hypothesis and from (i) and (iv) that $\mathcal{L}{\otimes }_{{\mathcal{O}}_Z}{\mathcal{O}}_Y$ is very ample for $p_2$; on the other hand, $\mathcal{L}={\Gamma }_f\ast (\mathcal{L}{\otimes }_{{\mathcal{O}}_Z}{\mathcal{O}}_Y)$ (I, 9.1.4), and ${\Gamma }_f$ is an immersion (I, 5.3.11); we may therefore apply (i bis).

To prove (i), we apply Definition (4.4.2) with $\mathcal{E}=\mathcal{L}$, noting that then $\mathbb{P}(\mathcal{E})$ is identified with $Y$ (4.1.4).

Let us prove (iii). There exists a quasi-coherent ${\mathcal{O}}_Y$-module $\mathcal{E}$ and a $Y$-immersion $i:X\to \mathbb{P}(\mathcal{E})=P$ such that $\mathcal{L}=i\ast ({\mathcal{O}}_P(1))$; setting ${\mathcal{E}}^{\prime }=g\ast (\mathcal{E})$, ${\mathcal{E}}^{\prime }$ is a quasi-coherent ${\mathcal{O}}_{Y^{\prime }}$-module, and $P^{\prime }=\mathbb{P}({\mathcal{E}}^{\prime })=P_{(Y^{\prime })}$ (4.1.3.1); $i_{(Y^{\prime })}$ is an immersion of $X_{(Y^{\prime })}$ into $P^{\prime }$ (I, 4.3.2), and ${\mathcal{L}}^{\prime }$ is isomorphic to $(i_{(Y^{\prime })})\ast ({\mathcal{O}}_{P^{\prime }}(1))$ (4.2.10).

To prove (iv), note that by hypothesis there is a $Y_i$-immersion $r_i:X_i\to P_i=\mathbb{P}({\mathcal{E}}_i)$, where ${\mathcal{E}}_i$ is a quasi-coherent ${\mathcal{O}}_{Y_i}$-module, and ${\mathcal{L}}_i=r_i\ast ({\mathcal{O}}_{P_i}(1))$ ($i=1,2$); $r_1{\times }_S{r}_2$ is an $S$-immersion of $X_1{\times }_S{X}_2$ into $P_1{\times }_S{P}_2$ (I, 4.3.1), and the inverse image of ${\mathcal{O}}_{P_1}(1){\otimes }_S{\mathcal{O}}_{P_2}(1)$ under this immersion is ${\mathcal{L}}_1{\otimes }_S{\mathcal{L}}_2$. Set $T=Y_1{\times }_S{Y}_2$, and let $p_1$, $p_2$ be the projections of $T$ to Y_1, Y_2 respectively. Setting $P_i^{\prime }=\mathbb{P}(p_i\ast ({\mathcal{E}}_i))$ ($i=1,2$), we have by (4.1.3.1) $P_i^{\prime }=P_i{\times }_{Y_i}T$, whence

  P_1' ×_T P_2' = (P_1 ×_{Y_1} T) ×_T (P_2 ×_{Y_2} T)
                = P_1 ×_{Y_1} (T ×_{Y_2} P_2)
                = P_1 ×_{Y_1} (Y_1 ×_S P_2)
                = P_1 ×_S P_2

up to canonical isomorphism. Similarly, ${\mathcal{O}}_{P_i^{\prime }}(1)={\mathcal{O}}_{P_i}(1){\otimes }_{Y_i}{\mathcal{O}}_T$ (4.1.3.2), and an analogous computation (based notably on (I, 9.1.9.1 and 9.1.2)) shows that, in the above identification, ${\mathcal{O}}_{P_1^{\prime }}(1){\otimes }_T{\mathcal{O}}_{P_2^{\prime }}(1)$ is identified with ${\mathcal{O}}_{P_1}(1){\otimes }_S{\mathcal{O}}_{P_2}(1)$. We may thus regard $r_1{\times }_S{r}_2$ as a $T$-immersion of $X_1{\times }_S{X}_2$ into $P_1^{\prime }{\times }_T{P}_2^{\prime }$, the inverse image of ${\mathcal{O}}_{P_1^{\prime }}(1){\otimes }_T{\mathcal{O}}_{P_2^{\prime }}(1)$ being ${\mathcal{L}}_1{\otimes }_S{\mathcal{L}}_2$. We finish the argument as in (4.4.8) using the Segre morphism.

It remains to prove (ii). We may first restrict to the case where $Z$ is an affine scheme, since in general there is a finite cover $(U_i)$ of $Z$ by affine opens; if the proposition is proved for $\mathcal{K}|g^{-1}(U_i)$, $\mathcal{L}|f^{-1}(g^{-1}(U_i))$, and an integer $n_i$, it suffices to take $n_0$ the largest of the $n_i$ to obtain the result for $\mathcal{K}$ and $\mathcal{L}$ (4.4.5). The hypothesis implies that $f$ and $g$ are separated, so $X$ and $Y$ are quasi-compact schemes.

There is an immersion $r:X\to P=\mathbb{P}(\mathcal{E})$, where $\mathcal{E}$ is a quasi-coherent ${\mathcal{O}}_Y$-module of finite type, and $\mathcal{L}=r\ast ({\mathcal{O}}_P(1))$, by (4.4.6). We shall see that there exists an ${\mathcal{O}}_P$-module $\mathcal{M}$ very ample relative to the composed morphism $P\stackrel{h}{\to }Y\stackrel{g}{\to }Z$ such that ${\mathcal{O}}_P(1)$ is isomorphic to $\mathcal{M}{\otimes }_Y{\mathcal{K}}^{\otimes (-m)}$ for some integer $m$. For $n\ge m+1$, ${\mathcal{O}}_P(1){\otimes }_Y{\mathcal{K}}^{\otimes n}$ will thus be very ample for $Z$ by hypothesis and (iv) applied to the morphisms $h:P\to Y$ and 1_Y; since $r$ is an immersion and $\mathcal{L}\otimes f\ast ({\mathcal{K}}^{\otimes n})=r\ast ({\mathcal{O}}_P(1){\otimes }_Y{\mathcal{K}}^{\otimes n})$, the conclusion will follow from (i bis). To prove our claim about ${\mathcal{O}}_P(1)$, we use the following lemma:

Lemma.

Let $Z$ be a quasi-compact scheme or a prescheme whose underlying space is Noetherian, $g:Y\to Z$ a quasi-compact morphism, $\mathcal{K}$ an invertible ${\mathcal{O}}_Y$-module very ample for $g$, and $\mathcal{E}$ a quasi-coherent ${\mathcal{O}}_Y$-module of finite type. Then there exists an integer $m_0$ such that, for every $m\ge m_0$, $\mathcal{E}$ is isomorphic to a quotient of an ${\mathcal{O}}_Y$-module of the form $g\ast (\mathcal{F})\otimes {\mathcal{K}}^{\otimes (-m)}$, where $\mathcal{F}$ is a quasi-coherent ${\mathcal{O}}_Z$-module of finite type (depending on $m$).

This lemma will be proved in (4.5.10.1); the reader may verify that (4.4.10) is not used anywhere in §4.5.

This lemma being granted, there is a closed immersion $j_1$ of $P$ into

$$P_1=\mathbb{P}(g\ast (\mathcal{F})\otimes {\mathcal{K}}^{\otimes (-m)})$$

such that ${\mathcal{O}}_P(1)$ is isomorphic to $j_1\ast ({\mathcal{O}}_{P_1}(1))$ (4.1.2). On the other hand, there is an isomorphism from P_1 to $P_2=\mathbb{P}(g\ast (\mathcal{F}))$, identifying ${\mathcal{O}}_{P_2}(1){\otimes }_Y{\mathcal{K}}^{\otimes (-m)}$ with ${\mathcal{O}}_{P_1}(1)$ (4.1.4); we therefore have a closed immersion $j_2:P\to P_2$ such that ${\mathcal{O}}_P(1)$ is isomorphic to $j_2\ast ({\mathcal{O}}_{P_2}(1)){\otimes }_Y{\mathcal{K}}^{\otimes (-m)}$. Finally, P_2 is identified with $P_3{\times }_{Z}Y$, where $P_3=\mathbb{P}(\mathcal{F})$, and ${\mathcal{O}}_{P_2}(1)$ with ${\mathcal{O}}_{P_3}(1){\otimes }_Z{\mathcal{O}}_Y$ (4.1.3). By definition, ${\mathcal{O}}_{P_3}(1)$ is very ample for $Z$; since so is $\mathcal{K}$, we conclude from (iv) that ${\mathcal{O}}_{P_2}(1){\otimes }_Y\mathcal{K}$ is very ample for $Z$; so is $\mathcal{M}=j_2\ast ({\mathcal{O}}_{P_2}(1){\otimes }_Y\mathcal{K})$ by (i bis), and ${\mathcal{O}}_P(1)$ is isomorphic to $\mathcal{M}{\otimes }_Y{\mathcal{K}}^{\otimes (-m-1)}$, which completes the proof.