§1. Affine morphisms

Most of the results in this section are the "global" counterparts of those in Chapter I, §1; they are therefore not essentially new and merely provide a convenient language for the sequel.

1.1. $S$-preschemes and ${\mathcal{O}}_S$-algebras

(1.1.1)

Let $S$ be a prescheme, $X$ an $S$-prescheme, and $f:X\to S$ its structure morphism. We know (0, 4.2.4) that the direct image $f_{\ast }({\mathcal{O}}_X)$ is an ${\mathcal{O}}_S$-algebra, which we

denote by $\mathcal{A}(X)$ when no confusion is likely. If $U$ is an open subset of $S$, then

$$\mathcal{A}(f^{-1}(U))=\mathcal{A}(X)|U.$$

Likewise, for every ${\mathcal{O}}_X$-module $\mathcal{F}$ (resp. every ${\mathcal{O}}_X$-algebra $\mathcal{B}$), we write $\mathcal{A}(\mathcal{F})$ (resp. $\mathcal{A}(\mathcal{B})$) for the direct image $f_{\ast }(\mathcal{F})$ (resp. $f_{\ast }(\mathcal{B})$), which is an $\mathcal{A}(X)$-module (resp. $\mathcal{A}(X)$-algebra), and not just an ${\mathcal{O}}_S$-module (resp. ${\mathcal{O}}_S$-algebra).

(1.1.2)

Let $Y$ be a second $S$-prescheme, $g:Y\to S$ its structure morphism, and $h:X\to Y$ an $S$-morphism, giving the commutative diagram

        h
   X ─────→ Y                                                            (1.1.2.1)
    \      /
   f \    / g
      ↘  ↙
       S

By definition $h=(\psi ,\theta )$, where $\theta :{\mathcal{O}}_Y\to h_{\ast }({\mathcal{O}}_X)={\psi }_{\ast }({\mathcal{O}}_X)$ is a homomorphism of sheaves of rings; we therefore obtain (0, 4.2.2) a homomorphism of ${\mathcal{O}}_S$-algebras $g_{\ast }(\theta ):g_{\ast }({\mathcal{O}}_Y)\to g_{\ast }(h_{\ast }({\mathcal{O}}_X))=f_{\ast }({\mathcal{O}}_X)$, in other words a homomorphism of ${\mathcal{O}}_S$-algebras $\mathcal{A}(Y)\to \mathcal{A}(X)$, which we also denote by $\mathcal{A}(h)$. If $h^{\prime }:Y\to Z$ is a second $S$-morphism, it is immediate that $\mathcal{A}(h^{\prime }\circ h)=\mathcal{A}(h)\circ \mathcal{A}(h^{\prime })$. We have thus defined a contravariant functor $\mathcal{A}(X)$ on the category of $S$-preschemes, with values in the category of ${\mathcal{O}}_S$-algebras.

Now let $\mathcal{F}$ be an ${\mathcal{O}}_X$-module, $\mathcal{G}$ an ${\mathcal{O}}_Y$-module, and $u:\mathcal{G}\to \mathcal{F}$ an $h$-morphism — that is (0, 4.4.1), a homomorphism of ${\mathcal{O}}_Y$-modules $\mathcal{G}\to h_{\ast }(\mathcal{F})$. Then $g_{\ast }(u):g_{\ast }(\mathcal{G})\to g_{\ast }(h_{\ast }(\mathcal{F}))=f_{\ast }(\mathcal{F})$ is a homomorphism $\mathcal{A}(\mathcal{G})\to \mathcal{A}(\mathcal{F})$ of ${\mathcal{O}}_S$-modules, which we denote by $\mathcal{A}(u)$; furthermore, the pair $(\mathcal{A}(h),\mathcal{A}(u))$ is a di-homomorphism from the $\mathcal{A}(Y)$-module $\mathcal{A}(\mathcal{G})$ to the $\mathcal{A}(X)$-module $\mathcal{A}(\mathcal{F})$.

(1.1.3)

If we fix the prescheme $S$, the pairs $(X,\mathcal{F})$ with $X$ an $S$-prescheme and $\mathcal{F}$ an ${\mathcal{O}}_X$-module form a category: a morphism $(X,\mathcal{F})\to (Y,\mathcal{G})$ is by definition a pair $(h,u)$ with $h:X\to Y$ an $S$-morphism and $u:\mathcal{G}\to \mathcal{F}$ an $h$-morphism. We may then say that $(\mathcal{A}(X),\mathcal{A}(\mathcal{F}))$ is a contravariant functor with values in the category whose objects are pairs consisting of an ${\mathcal{O}}_S$-algebra and a module over that algebra, and whose morphisms are di-homomorphisms.

1.2. Affine preschemes over a prescheme

Definition.

Let $X$ be an $S$-prescheme and $f:X\to S$ its structure morphism. We say that $X$ is affine over $S$ if there exists a covering $(S_{\alpha })$ of $S$ by affine opens such that for every $\alpha$, the prescheme induced by $X$ on $f^{-1}(S_{\alpha })$ is affine.

Example.

Every closed subprescheme of $S$ is an $S$-prescheme affine over $S$ (I, 4.2.3 and 4.2.4).

Remark.

A prescheme $X$ affine over $S$ need not itself be an affine scheme, as the example $X=S$ from (1.2.2) shows. On the other hand, if an affine scheme $X$ is an $S$-prescheme, $X$ is not necessarily affine over $S$ (see (1.3.3) below). Recall

however that if $S$ is a scheme, then every $S$-prescheme that is an affine scheme is affine over $S$ (I, 5.5.10).

Proposition.

Every $S$-prescheme affine over $S$ is separated over $S$ (in other words, it is an $S$-scheme).

Proof. This follows immediately from (I, 5.5.5) and (I, 5.5.8).

Proposition.

Let $X$ be an $S$-scheme affine over $S$ and $f:X\to S$ its structure morphism. For every open $U\subset S$, $f^{-1}(U)$ is affine over $U$.

Proof. By Definition (1.2.1), we reduce to the case where $S=\mathrm{Spec}(A)$ and $X=\mathrm{Spec}(B)$ are affine, with $f=({}^a\varphi ,\tilde{\varphi })$ and $\varphi :A\to B$ a homomorphism. Since the $D(g)$ for $g\in A$ form a basis of $S$, we reduce to $U=D(g)$; but $f^{-1}(U)=D(\varphi (g))$ (I, 1.2.2.2), which gives the proposition.

Proposition.

Let $X$ be an $S$-scheme affine over $S$ and $f:X\to S$ its structure morphism. For every quasi-coherent ${\mathcal{O}}_X$-module $\mathcal{F}$, $f_{\ast }(\mathcal{F})$ is a quasi-coherent ${\mathcal{O}}_S$-module.

Proof. Given (1.2.4), this follows from (I, 9.2.2 a).

In particular, the ${\mathcal{O}}_S$-algebra $\mathcal{A}(X)=f_{\ast }({\mathcal{O}}_X)$ is quasi-coherent.

Proposition.

Let $X$ be an $S$-scheme affine over $S$. For every $S$-prescheme $Y$, the map $h\mapsto \mathcal{A}(h)$ from ${\mathrm{Hom}}_S(Y,X)$ to $\mathrm{Hom}(\mathcal{A}(X),\mathcal{A}(Y))$ (1.1.2) is bijective.

Proof. Let $f:X\to S$ and $g:Y\to S$ be the structure morphisms.

First, suppose $S=\mathrm{Spec}(A)$ and $X=\mathrm{Spec}(B)$ are affine. We must show that every homomorphism $\omega :f_{\ast }({\mathcal{O}}_X)\to g_{\ast }({\mathcal{O}}_Y)$ of ${\mathcal{O}}_S$-algebras arises from a unique $S$-morphism $h:Y\to X$ via $\mathcal{A}(h)=\omega$. By definition, for every open $U\subset S$, $\omega$ defines a homomorphism

  ω_U = Γ(U, ω) : Γ(f⁻¹(U), 𝒪_X) → Γ(g⁻¹(U), 𝒪_Y)

of $\Gamma (U,{\mathcal{O}}_S)$-algebras. In particular, taking $U=S$, we obtain a homomorphism $\varphi :\Gamma (X,{\mathcal{O}}_X)\to \Gamma (Y,{\mathcal{O}}_Y)$ of $\Gamma (S,{\mathcal{O}}_S)$-algebras, to which corresponds a well-defined $S$-morphism $h:Y\to X$ since $X$ is affine (I, 2.2.4). It remains to prove $\mathcal{A}(h)=\omega$, that is: for every open $U$ in a basis of $S$, ${\omega }_U$ coincides with the algebra homomorphism ${\varphi }_U$ corresponding to the restriction $g^{-1}(U)\to f^{-1}(U)$ of $h$. We may take $U=D(\lambda )$ with $\lambda \in S$; then if $f=({}^a\rho ,\tilde{\rho })$ with $\rho :A\to B$ a ring homomorphism, $f^{-1}(U)=D(\mu )$ for $\mu =\rho (\lambda )$, and $\Gamma (f^{-1}(U),{\mathcal{O}}_X)$ is the ring of fractions $B_{\mu }$. The diagram

   B ──────φ─────→ Γ(Y, 𝒪_Y)
   │                  │
   ↓                  ↓
   B_μ ────φ_U───→ Γ(g⁻¹(U), 𝒪_Y)

commutes, and so does the analogous diagram with ${\varphi }_U$ replaced by ${\omega }_U$; the equality ${\varphi }_U={\omega }_U$ then follows from the universal property of rings of fractions (0, 1.2.4).

For the general case, let $(S_{\alpha })$ be a covering of $S$ by affine opens

such that the $f^{-1}(S_{\alpha })$ are affine. Every homomorphism $\omega :\mathcal{A}(X)\to \mathcal{A}(Y)$ of ${\mathcal{O}}_S$-algebras restricts to a family of homomorphisms

$${\omega }_{\alpha }:\mathcal{A}(f^{-1}(S_{\alpha }))\to \mathcal{A}(g^{-1}(S_{\alpha }))$$

of ${\mathcal{O}}_{S_{\alpha }}$-algebras, hence to a family of $S_{\alpha }$-morphisms $h_{\alpha }:g^{-1}(S_{\alpha })\to f^{-1}(S_{\alpha })$ by the affine case above. We need only check that on every affine open $U$ in a basis of $S_{\alpha }\cap S_{\beta }$, the restrictions of $h_{\alpha }$ and $h_{\beta }$ to $g^{-1}(U)$ agree, which is evident since both correspond to the restriction $\mathcal{A}(X)|U\to \mathcal{A}(Y)|U$ of $\omega$.

Corollary.

Let $X$, $Y$ be two $S$-schemes affine over $S$. An $S$-morphism $h:Y\to X$ is an isomorphism if and only if $\mathcal{A}(h):\mathcal{A}(X)\to \mathcal{A}(Y)$ is one.

Proof. Immediate from (1.2.7) and the functoriality of $\mathcal{A}(X)$.

1.3. Affine prescheme over $S$ associated to an ${\mathcal{O}}_S$-algebra

Proposition.

Let $S$ be a prescheme. For every quasi-coherent ${\mathcal{O}}_S$-algebra $\mathcal{B}$, there exists a prescheme $X$ affine over $S$, defined up to unique $S$-isomorphism, such that $\mathcal{A}(X)=\mathcal{B}$.

Proof. Uniqueness follows from (1.2.8); we prove existence. For every affine open $U\subset S$, set $X_U=\mathrm{Spec}(\Gamma (U,\mathcal{B}))$; since $\Gamma (U,\mathcal{B})$ is a $\Gamma (U,{\mathcal{O}}_S)$-algebra, X_U is an $S$-prescheme (I, 1.6.1). As $\mathcal{B}$ is quasi-coherent, the ${\mathcal{O}}_S$-algebra $\mathcal{A}(X_U)$ is canonically identified with $\mathcal{B}|U$ (I, 1.3.7, 1.3.13, 1.6.3). For a second affine open $V\subset S$, let $X_{U,V}$ be the prescheme induced by X_U on $f_U^{-1}(U\cap V)$, where $f_U:X_U\to S$ is the structure morphism; then $X_{U,V}$ and $X_{V,U}$ are affine over $U\cap V$ (1.2.5), and by definition $\mathcal{A}(X_{U,V})$ and $\mathcal{A}(X_{V,U})$ both identify canonically with $\mathcal{B}|(U\cap V)$. Hence (1.2.8) there is a canonical $S$-isomorphism ${\theta }_{U,V}:X_{V,U}\to X_{U,V}$; furthermore, if $W$ is a third affine open of $S$ and ${\theta }_{U,V}^{\prime }$, ${\theta }_{V,W}^{\prime }$, ${\theta }_{U,W}^{\prime }$ are the restrictions to the inverse images of $U\cap V\cap W$, then ${\theta }_{U,V}^{\prime }\circ {\theta }_{V,W}^{\prime }={\theta }_{U,W}^{\prime }$. Consequently there exists a prescheme $X$, a covering (T_U) of $X$ by affine opens, and for each $U$ an isomorphism ${\varphi }_U:X_U\to T_U$ such that ${\varphi }_U$ carries $f_U^{-1}(U\cap V)$ onto $T_U\cap T_V$ with ${\theta }_{U,V}={\varphi }_U^{-1}\circ {\varphi }_V$ (I, 2.3.1). The morphism $g_U=f_U\circ {\varphi }_U^{-1}$ makes T_U an $S$-prescheme, and $g_U$, $g_V$ coincide on $T_U\cap T_V$, so $X$ is an $S$-prescheme. It is then clear by construction that $X$ is affine over $S$ and that $\mathcal{A}(T_U)=\mathcal{B}|U$, whence $\mathcal{A}(X)=\mathcal{B}$.

We say that the $S$-scheme $X$ so defined is associated to the ${\mathcal{O}}_S$-algebra $\mathcal{B}$, or is the spectrum of $\mathcal{B}$, and we denote it by $\mathrm{Spec}(\mathcal{B})$.

Corollary.

Let $X$ be a prescheme affine over $S$ and $f:X\to S$ the structure morphism. For every affine open $U\subset S$, the prescheme induced on $f^{-1}(U)$ is the affine scheme with ring $\Gamma (U,\mathcal{A}(X))$.

Proof. Since by (1.2.6) and (1.3.1) we may take $X$ to be associated to an ${\mathcal{O}}_S$-algebra, the corollary follows from the construction of $X$ in (1.3.1).

Example.

Let $S$ be the affine plane over a field $K$ with the origin doubled (I, 5.5.11); with the notation of (I, 5.5.11), $S$ is the union of two affine opens Y_1, Y_2. If $f$ is the open immersion $Y_1\to S$, then $f^{-1}(Y_2)=Y_1\cap Y_2$ is not an affine open in Y_1 (loc. cit.), giving an example of an affine scheme that is not affine over $S$.

Corollary.

Let $S$ be an affine scheme. An $S$-prescheme $X$ is affine over $S$ if and only if $X$ is an affine scheme.

Corollary.

Let $X$ be a prescheme affine over a prescheme $S$, and let $Y$ be an $X$-prescheme. Then $Y$ is affine over $S$ if and only if $Y$ is affine over $X$.

Proof. We reduce immediately to the case where $S$ is an affine scheme, and then, by (1.3.4), so is $X$; the two conditions both say that $Y$ is an affine scheme (1.3.4).

(1.3.6)

Let $X$ be a prescheme affine over $S$. To give a prescheme $Y$ affine over $X$ is, by Corollary (1.3.5), the same as to give a prescheme $Y$ affine over $S$ together with an $S$-morphism $g:Y\to X$; in other words ((1.3.1) and (1.2.7)), it is the same as to give a quasi-coherent ${\mathcal{O}}_S$-algebra $\mathcal{B}$ together with a homomorphism $\mathcal{A}(X)\to \mathcal{B}$ of ${\mathcal{O}}_S$-algebras (which may be viewed as defining on $\mathcal{B}$ an $\mathcal{A}(X)$-algebra structure). If $f:X\to S$ is the structure morphism, then $\mathcal{B}=f_{\ast }(g_{\ast }({\mathcal{O}}_Y))$.

Corollary.

Let $X$ be a prescheme affine over $S$. Then $X$ is of finite type over $S$ if and only if the quasi-coherent ${\mathcal{O}}_S$-algebra $\mathcal{A}(X)$ is of finite type (I, 9.6.2).

Proof. By definition (I, 9.6.2) we reduce to the case where $S$ is affine; then $X$ is affine (1.3.4), and if $S=\mathrm{Spec}(A)$, $X=\mathrm{Spec}(B)$, then $\mathcal{A}(X)$ is the ${\mathcal{O}}_S$-algebra $B$. Since $\Gamma (S,B)=B$, the corollary follows from (I, 9.6.2) and (I, 6.3.3).

Corollary.

Let $X$ be a prescheme affine over $S$. Then $X$ is reduced if and only if the quasi-coherent ${\mathcal{O}}_S$-algebra $\mathcal{A}(X)$ is reduced (0, 4.1.4).

Proof. The question is local on $S$; by Corollary (1.3.2), the assertion follows from (I, 5.1.1) and (I, 5.1.4).

1.4. Quasi-coherent sheaves on an affine prescheme over $S$

Proposition.

Let $X$ be a prescheme affine over $S$, $Y$ an $S$-prescheme, and $\mathcal{F}$ (resp. $\mathcal{G}$) a quasi-coherent ${\mathcal{O}}_X$-module (resp. ${\mathcal{O}}_Y$-module). Then the map $(h,u)\mapsto (\mathcal{A}(h),\mathcal{A}(u))$ from the set of morphisms $(Y,\mathcal{G})\to (X,\mathcal{F})$ to the set of di-homomorphisms $(\mathcal{A}(X),\mathcal{A}(\mathcal{F}))\to (\mathcal{A}(Y),\mathcal{A}(\mathcal{G}))$ ((1.1.2) and (1.1.3)) is bijective.

Proof. The proof follows exactly the same lines as that of (1.2.7), using (I, 2.2.5) and (I, 2.2.4); details are left to the reader.

Corollary.

If, in addition to the hypotheses of (1.4.1), $Y$ is also affine over $S$, then $(h,u)$ is an isomorphism if and only if $(\mathcal{A}(h),\mathcal{A}(u))$ is a di-isomorphism.

Proposition.

For every pair $(\mathcal{B},\mathcal{M})$ consisting of a quasi-coherent ${\mathcal{O}}_S$-algebra $\mathcal{B}$ and a quasi-coherent $\mathcal{B}$-module $\mathcal{M}$ (viewed as an ${\mathcal{O}}_S$-module or as a $\mathcal{B}$-module — these are equivalent (I, 9.6.1)), there exists a pair $(X,\mathcal{F})$ consisting of a prescheme $X$ affine over $S$ and a quasi-coherent ${\mathcal{O}}_X$-module $\mathcal{F}$ such that $\mathcal{A}(X)=\mathcal{B}$ and $\mathcal{A}(\mathcal{F})=\mathcal{M}$; this pair is determined up to unique isomorphism.

Proof. Uniqueness follows from (1.4.1) and (1.4.2); existence is proved as in (1.3.1), and we again leave details to the reader.

We denote by $\mathcal{M}$ the ${\mathcal{O}}_X$-module $\mathcal{F}$ so obtained, and call it associated to the quasi-coherent $\mathcal{B}$-module $\mathcal{M}$. For every affine open $U\subset S$, $\mathcal{F}|p^{-1}(U)$ (where $p$ is the structure morphism $X\to S$) is canonically isomorphic to $\Gamma (U,\mathcal{M})$.

Corollary.

On the category of quasi-coherent $\mathcal{B}$-modules, $\tilde{\mathcal{M}}$ is a covariant additive exact functor in $\mathcal{M}$ that commutes with direct limits and direct sums.

Proof. We reduce immediately to the case where $S$ is affine; the corollary then follows from (I, 1.3.5), (I, 1.3.9), and (I, 1.3.11).

Corollary.

Under the hypotheses of (1.4.3), $\tilde{\mathcal{M}}$ is an ${\mathcal{O}}_X$-module of finite type if and only if $\mathcal{M}$ is a $\mathcal{B}$-module of finite type.

Proof. We reduce to $S=\mathrm{Spec}(A)$ affine. Then $\mathcal{B}=B$, where $B$ is an $A$-algebra of finite type (I, 9.6.2), and $\mathcal{M}=M$, where $M$ is a $B$-module (I, 1.3.13). On the prescheme $X$, ${\mathcal{O}}_X$ is associated to $B$ and $\mathcal{M}$ to the $B$-module $M$; so $\mathcal{M}$ is of finite type iff $M$ is of finite type (I, 1.3.13), whence the claim.

Proposition.

Let $Y$ be a prescheme affine over $S$, and let $X$, $X^{\prime }$ be two preschemes affine over $Y$ (hence also over $S$ by (1.3.5)). Set $\mathcal{B}=\mathcal{A}(Y)$, $\mathcal{A}=\mathcal{A}(X)$, ${\mathcal{A}}^{\prime }=\mathcal{A}(X^{\prime })$. Then $X{\times }_Y{X}^{\prime }$ is affine over $Y$ (and hence also over $S$), and $\mathcal{A}(X{\times }_Y{X}^{\prime })$ is canonically identified with $\mathcal{A}{\otimes }_{\mathcal{B}}{\mathcal{A}}^{\prime }$.

Proof. By (I, 9.6.1), $\mathcal{A}{\otimes }_{\mathcal{B}}{\mathcal{A}}^{\prime }$ is a quasi-coherent $\mathcal{B}$-algebra, and thus also a quasi-coherent ${\mathcal{O}}_S$-algebra (I, 9.6.1). Let $Z$ be the spectrum of $\mathcal{A}{\otimes }_{\mathcal{B}}{\mathcal{A}}^{\prime }$ (1.3.1). The canonical $\mathcal{B}$-homomorphisms $\mathcal{A}\to \mathcal{A}{\otimes }_{\mathcal{B}}{\mathcal{A}}^{\prime }$ and ${\mathcal{A}}^{\prime }\to \mathcal{A}{\otimes }_{\mathcal{B}}{\mathcal{A}}^{\prime }$ correspond by (1.2.7) to $Y$-morphisms $p:Z\to X$ and $p^{\prime }:Z\to X^{\prime }$. To see that (Z, p, p') is a product $X{\times }_Y{X}^{\prime }$, we may reduce to the case where $S$ is an affine scheme with ring $C$ (I, 3.2.6.4). Then $Y$, $X$, $X^{\prime }$ are affine schemes (1.3.4) with rings $B$, $A$, $A^{\prime }$ which are $C$-algebras such that $\mathcal{B}=B$, $\mathcal{A}=A$, ${\mathcal{A}}^{\prime }=A^{\prime }$. Then (I, 1.3.13) gives $\mathcal{A}{\otimes }_{\mathcal{B}}{\mathcal{A}}^{\prime }=A{\otimes }_B{A}^{\prime }$, so the ring of $Z$ is $A{\otimes }_B{A}^{\prime }$ and the morphisms $p$, $p^{\prime }$ correspond to the canonical homomorphisms $A\to A{\otimes }_B{A}^{\prime }$ and $A^{\prime }\to A{\otimes }_B{A}^{\prime }$. The proposition follows from (I, 3.2.2).

Corollary.

Let $\mathcal{F}$ (resp. ${\mathcal{F}}^{\prime }$) be a quasi-coherent ${\mathcal{O}}_X$-module (resp. ${\mathcal{O}}_{X^{\prime }}$-module). Then $\mathcal{A}(\mathcal{F}{\otimes }_Y{\mathcal{F}}^{\prime })$ is canonically identified with $\mathcal{A}(\mathcal{F}){\otimes }_{\mathcal{A}(Y)}\mathcal{A}({\mathcal{F}}^{\prime })$.

Proof. We know (I, 9.1.2) that $\mathcal{F}{\otimes }_Y{\mathcal{F}}^{\prime }$ is quasi-coherent on $X{\times }_Y{X}^{\prime }$. Let $g:Y\to S$, $f:X\to Y$, $f^{\prime }:X^{\prime }\to Y$ be the structure morphisms, so that the structure morphism

$h:Z\to S$ equals $g\circ f\circ p=g\circ f^{\prime }\circ p^{\prime }$. We define a canonical homomorphism

  𝒜(ℱ) ⊗_{𝒜(Y)} 𝒜(ℱ') → 𝒜(ℱ ⊗_Y ℱ')

as follows: for every open $U\subset S$, the canonical homomorphisms $\Gamma (f^{-1}(g^{-1}(U)),\mathcal{F})\to \Gamma (h^{-1}(U),p\ast (\mathcal{F}))$ and $\Gamma (f^{\prime -1}(g^{-1}(U)),{\mathcal{F}}^{\prime })\to \Gamma (h^{-1}(U),p^{\prime }\ast ({\mathcal{F}}^{\prime }))$ (0, 4.4.3) give a canonical homomorphism

  Γ(f⁻¹(g⁻¹(U)), ℱ) ⊗_{Γ(g⁻¹(U), 𝒪_Y)} Γ(f'⁻¹(g⁻¹(U)), ℱ')
    → Γ(h⁻¹(U), p*(ℱ)) ⊗_{Γ(h⁻¹(U), 𝒪_Z)} Γ(h⁻¹(U), p'*(ℱ')).

To see this is an isomorphism of $\mathcal{A}(Z)$-modules, we reduce to the affine case: $S$ (and hence $X$, $X^{\prime }$, $Y$, $X{\times }_Y{X}^{\prime }$) affine, and (with the notation of (1.4.6)) $\mathcal{F}=M$, ${\mathcal{F}}^{\prime }=M^{\prime }$ with $M$ an $A$-module and $M^{\prime }$ an $A^{\prime }$-module. Then $\mathcal{F}{\otimes }_Y{\mathcal{F}}^{\prime }$ identifies with the sheaf on $X{\times }_Y{X}^{\prime }$ associated to the $(A{\otimes }_B{A}^{\prime })$-module $M{\otimes }_B{M}^{\prime }$ (I, 9.1.3), and the corollary follows from the canonical identification of ${\mathcal{O}}_S$-modules $M{\otimes }_B{M}^{\prime }$ and $M{\otimes }_B{M}^{\prime }$ (where $M$, $M^{\prime }$, $B$ are viewed as $C$-modules) (I, 1.3.13 and 1.6.3).

Applying (1.4.7) in the special case $X=Y$, ${\mathcal{F}}^{\prime }={\mathcal{O}}_{X^{\prime }}$, we see that the ${\mathcal{A}}^{\prime }$-module $\mathcal{A}(f^{\prime }\ast (\mathcal{F}))$ identifies with $\mathcal{A}(\mathcal{F}){\otimes }_{\mathcal{B}}{\mathcal{A}}^{\prime }$.

(1.4.8)

In particular, when $X=X^{\prime }=Y$ (with $X$ affine over $S$), for any two quasi-coherent ${\mathcal{O}}_X$-modules $\mathcal{F}$, $\mathcal{G}$,

  𝒜(ℱ ⊗_{𝒪_X} 𝒢) = 𝒜(ℱ) ⊗_{𝒜(X)} 𝒜(𝒢)                                    (1.4.8.1)

up to canonical functorial isomorphism. If furthermore $\mathcal{F}$ admits a finite presentation, then (I, 1.6.3) and (I, 1.3.12) give

  𝒜(𝓗𝓸𝓶_X(ℱ, 𝒢)) = 𝓗𝓸𝓶_{𝒜(X)}(𝒜(ℱ), 𝒜(𝒢))                                  (1.4.8.2)

up to canonical isomorphism.

Remark.

If $X$, $X^{\prime }$ are two preschemes affine over $S$, then the disjoint sum $X\bigsqcup X^{\prime }$ is also affine over $S$, since the sum of two affine schemes is affine.

Proposition.

Let $S$ be a prescheme, $\mathcal{B}$ a quasi-coherent ${\mathcal{O}}_S$-algebra, and $X=\mathrm{Spec}(\mathcal{B})$. For every quasi-coherent sheaf of ideals $\mathcal{J}$ of $\mathcal{B}$, $\tilde{\mathcal{J}}$ is a quasi-coherent sheaf of ideals of ${\mathcal{O}}_X$, and the closed subprescheme $Y$ of $X$ it defines is canonically isomorphic to $\mathrm{Spec}(\mathcal{B}/\mathcal{J})$.

Proof. By (I, 4.2.3), $Y$ is affine over $S$; by (1.3.1) we reduce to $S$ affine, and the assertion is then (I, 4.1.2).

The conclusion of (1.4.10) can be restated: if $h:\mathcal{B}\to {\mathcal{B}}^{\prime }$ is a surjective homomorphism of quasi-coherent ${\mathcal{O}}_S$-algebras, then Spec(h) : Spec(ℬ') → Spec(ℬ) is a closed immersion.

Proposition.

Let $S$ be a prescheme, $\mathcal{B}$ a quasi-coherent ${\mathcal{O}}_S$-algebra, and $X=\mathrm{Spec}(\mathcal{B})$. For every quasi-coherent sheaf of ideals $\mathcal{K}$ of ${\mathcal{O}}_S$ (denoting by $f:X\to S$ the structure morphism), $f\ast (\mathcal{K}){\mathcal{O}}_X=\widetilde{\mathcal{K}\mathcal{B}}$ up to canonical isomorphism.

Proof. The question is local on $S$, so we reduce to $S=\mathrm{Spec}(A)$ affine, where the assertion is just (I, 1.6.9).

1.5. Change of base prescheme

Proposition.

Let $X$ be a prescheme affine over $S$. For every base change $g:S^{\prime }\to S$, $X^{\prime }=X_{(S^{\prime })}=X{\times }_S{S}^{\prime }$ is affine over $S^{\prime }$.

Proof. If $f^{\prime }:X^{\prime }\to S^{\prime }$ is the projection, it suffices to prove that $f^{\prime -1}(U^{\prime })$ is an affine open for every affine open $U^{\prime }\subset S^{\prime }$ such that $g(U^{\prime })$ is contained in an affine open $U\subset S$ (1.2.1). We may thus assume $S$ and $S^{\prime }$ are affine; it then suffices to show $X^{\prime }$ is affine (1.3.4). But then $X$ is affine, and if $A$, $A^{\prime }$, $B$ are the rings of $S$, $S^{\prime }$, $X$, then $X^{\prime }$ is the affine scheme with ring $A^{\prime }{\otimes }_{A}B$ (I, 3.2.2).

Corollary.

Under the hypotheses of (1.5.1), let $f:X\to S$ be the structure morphism and $f^{\prime }:X^{\prime }\to S^{\prime }$, $g^{\prime }:X^{\prime }\to X$ the projections, so that the diagram

   X ←─g'── X'
   │        │
 f │        │ f'
   ↓        ↓
   S ←──g── S'

is commutative. For every quasi-coherent ${\mathcal{O}}_X$-module $\mathcal{F}$, there is a canonical isomorphism of ${\mathcal{O}}_{S^{\prime }}$-modules

$$u:g\ast (f_{\ast }(\mathcal{F}))\stackrel{\sim }{\to }{f}_{\ast }^{\prime }(g^{\prime }\ast (\mathcal{F})).(\mathrm{1.5.2.1})$$

In particular, there is a canonical isomorphism from $\mathcal{A}(X^{\prime })$ to $g\ast (\mathcal{A}(X))$.

Proof. To define $u$, it suffices to define a homomorphism

  v : f_*(ℱ) → g_*(f'_*(g'*(ℱ))) = f_*(g'_*(g'*(ℱ)))

and set $u=v♯$ (0, 4.4.3). Take $v=f_{\ast }(\rho )$ with $\rho :\mathcal{F}\to g_{\ast }^{\prime }(g^{\prime }\ast (\mathcal{F}))$ the canonical homomorphism (0, 4.4.3). To prove that $u$ is an isomorphism, we may assume $S$, $S^{\prime }$ (hence $X$, $X^{\prime }$) are affine. With the notation of (1.5.1), $\mathcal{F}=\tilde{M}$ for a $B$-module $M$, and one checks that both $g\ast (f_{\ast }(\mathcal{F}))$ and $f_{\ast }^{\prime }(g^{\prime }\ast (\mathcal{F}))$ equal the ${\mathcal{O}}_{S^{\prime }}$-module associated to the $A^{\prime }$-module $A^{\prime }{\otimes }_{A}M$ (with $M$ viewed as an $A$-module), and that $u$ corresponds to the identity (I, 1.6.3, 1.6.5, 1.6.7).

Remark.

One should not expect (1.5.2) to remain valid when $X$ is no longer affine over $S$ — not even when $S^{\prime }=\mathrm{Spec}(\kappa (s))$ for $s\in S$ with $S^{\prime }\to S$ the canonical morphism (I, 2.4.5), in which case $X^{\prime }$ is just the fibre $f^{-1}(s)$ (I, 3.6.2). In other words, when $X$ is not affine over $S$, the operation

"direct image of quasi-coherent sheaves" does not commute with the operation "passage to fibres". We will see however in Chapter III (III, 4.2.4) a result in this direction, of an "asymptotic" character, valid for coherent sheaves on $X$ when $f$ is proper (5.4) and $S$ is Noetherian.

Corollary.

For every prescheme $X$ affine over $S$ and every $s\in S$, the fibre $f^{-1}(s)$ (where $f:X\to S$ is the structure morphism) is an affine scheme.

Proof. Apply (1.5.1) with $S^{\prime }=\mathrm{Spec}(\kappa (s))$ and use (1.3.4).

Corollary.

Let $X$ be an $S$-prescheme and $S^{\prime }$ a prescheme affine over $S$. Then $X^{\prime }=X_{(S^{\prime })}$ is a prescheme affine over $X$. Furthermore, if $f:X\to S$ is the structure morphism, there is a canonical isomorphism of ${\mathcal{O}}_X$-algebras $\mathcal{A}(X^{\prime })\stackrel{\sim }{\to }f\ast (\mathcal{A}(S^{\prime }))$, and for every quasi-coherent $\mathcal{A}(S^{\prime })$-module $\mathcal{M}$ a canonical di-isomorphism $f\ast (\mathcal{M})\stackrel{\sim }{\to }\mathcal{A}(f^{\prime }\ast (\tilde{\mathcal{M}}))$, where $f^{\prime }=f_{(S^{\prime })}:X^{\prime }\to S^{\prime }$ is the structure morphism.

Proof. Swap the roles of $X$ and $S^{\prime }$ in (1.5.1) and (1.5.2).

(1.5.6)

Now let $S$, $S^{\prime }$ be two preschemes, $q:S^{\prime }\to S$ a morphism, $\mathcal{B}$ (resp. ${\mathcal{B}}^{\prime }$) a quasi-coherent ${\mathcal{O}}_S$-algebra (resp. ${\mathcal{O}}_{S^{\prime }}$-algebra), and $u:\mathcal{B}\to {\mathcal{B}}^{\prime }$ a $q$-morphism — that is, a homomorphism $\mathcal{B}\to q_{\ast }({\mathcal{B}}^{\prime })$ of ${\mathcal{O}}_S$-algebras. If $X=\mathrm{Spec}(\mathcal{B})$ and $X^{\prime }=\mathrm{Spec}({\mathcal{B}}^{\prime })$, we obtain canonically a morphism

  v = Spec(u) : X' → X

such that the diagram

   X' ──v──→ X
   │         │                                                            (1.5.6.1)
   ↓         ↓
   S' ──q──→ S

commutes (the vertical arrows being the structure morphisms). Indeed, the datum of $u$ is equivalent to that of a homomorphism $u♯:q\ast (\mathcal{B})\to {\mathcal{B}}^{\prime }$ of quasi-coherent ${\mathcal{O}}_{S^{\prime }}$-algebras (0, 4.4.3), which canonically defines an $S^{\prime }$-morphism

$$w:\mathrm{Spec}({\mathcal{B}}^{\prime })\to \mathrm{Spec}(q\ast (\mathcal{B}))$$

with $\mathcal{A}(w)=u♯$ (1.2.7). On the other hand, (1.5.2) gives a canonical identification $\mathrm{Spec}(q\ast (\mathcal{B}))\cong X{\times }_S{S}^{\prime }$, and $v$ is the composition $X^{\prime }\stackrel{w}{\to }X{\times }_S{S}^{\prime }\stackrel{p{r}_1}{\to }X$; the commutativity of (1.5.6.1) follows from the definitions. Let $U$ (resp. $U^{\prime }$) be an affine open of $S$ (resp. $S^{\prime }$) with $q(U^{\prime })\subset U$, with rings $A=\Gamma (U,{\mathcal{O}}_S)$, $A^{\prime }=\Gamma (U^{\prime },{\mathcal{O}}_{S^{\prime }})$, and $B=\Gamma (U,\mathcal{B})$, $B^{\prime }=\Gamma (U^{\prime },{\mathcal{B}}^{\prime })$. The restriction of $u$ to a $(q|U^{\prime })$-morphism $\mathcal{B}|U\to {\mathcal{B}}^{\prime }|U^{\prime }$ corresponds to a di-homomorphism of algebras $B\to B^{\prime }$; if $V$, $V^{\prime }$ are the inverse images of $U$, $U^{\prime }$ in $X$, $X^{\prime }$ under the structure morphisms, the morphism $V^{\prime }\to V$ (the restriction of $v$) corresponds (I, 1.7.3) to this di-homomorphism.

(1.5.7)

Under the same hypotheses as (1.5.6), let $\mathcal{M}$ be a quasi-coherent $\mathcal{B}$-module. There is then a canonical isomorphism of ${\mathcal{O}}_{X^{\prime }}$-modules

  v*(ℳ̃) ⥲ (q*(ℳ) ⊗_{q*(ℬ)} ℬ')̃.                                          (1.5.7.1)

Indeed, the canonical isomorphism (1.5.2.1) gives a canonical identification of $p{r}_1\ast (\tilde{\mathcal{M}})$ with the sheaf on $\mathrm{Spec}(q\ast (\mathcal{B}))$ associated to the $q\ast (\mathcal{B})$-module $q\ast (\mathcal{M})$, and one applies (1.4.7).

1.6. Affine morphisms

(1.6.1)

We say that a morphism $f:X\to Y$ of preschemes is affine if it makes $X$ an affine prescheme over $Y$. The properties of preschemes affine over another translate as follows in this language:

Proposition.

1. A closed immersion is affine.
2. The composition of two affine morphisms is affine.
3. If $f:X\to Y$ is an affine $S$-morphism, then $f_{(S^{\prime })}:X_{(S^{\prime })}\to Y_{(S^{\prime })}$ is affine for every base change $S^{\prime }\to S$.
4. If $f:X\to Y$ and $f^{\prime }:X^{\prime }\to Y^{\prime }$ are two affine $S$-morphisms, then $f{\times }_S{f}^{\prime }:X{\times }_S{X}^{\prime }\to Y{\times }_S{Y}^{\prime }$ is affine.
5. If $f:X\to Y$ and $g:Y\to Z$ are two morphisms such that $g\circ f$ is affine and $g$ is separated, then $f$ is affine.
6. If $f$ is affine, so is $f_{red}$.

Proof. By (I, 5.5.12), it suffices to prove (i), (ii), (iii). But (i) is just Example (1.2.2), (ii) is just Corollary (1.3.5), and (iii) follows from (1.5.1) since $X_{(S^{\prime })}$ identifies with $X{\times }_Y{Y}_{(S^{\prime })}$ (I, 3.3.11).

Corollary.

If $X$ is an affine scheme and $Y$ is a scheme, then every morphism $f:X\to Y$ is affine.

Proposition.

Let $Y$ be a locally Noetherian prescheme and $f:X\to Y$ a morphism of finite type. Then $f$ is affine if and only if $f_{red}$ is.

Proof. By (1.6.2)(vi), it suffices to prove sufficiency. We may suppose $Y$ affine and Noetherian, and must show $X$ is affine; then $Y_{red}$ is affine, so by hypothesis $X_{red}$ is affine. Since $X$ is Noetherian, the conclusion follows from (I, 6.1.7).

1.7. Vector bundle associated to a sheaf of modules

(1.7.1)

Let $A$ be a ring and $E$ an $A$-module. Recall that the symmetric algebra on $E$, denoted $\mathbb{S}(E)$ (or ${\mathbb{S}}_A(E)$), is the quotient of the tensor algebra $\mathbb{T}(E)$ by the two-sided ideal generated by $x\otimes y-y\otimes x$ for $x,y\in E$. The algebra $\mathbb{S}(E)$ is characterized by the following universal property: if $\sigma :E\to \mathbb{S}(E)$ is the canonical map (obtained by composing $E\to \mathbb{T}(E)$ with $\mathbb{T}(E)\to \mathbb{S}(E)$), then every $A$-linear map $E\to B$ with $B$ a commutative $A$-algebra factors uniquely as $E\stackrel{\sigma }{\to }\mathbb{S}(E)\stackrel{g}{\to }B$, where $g$ is an $A$-homomorphism of algebras. From this characterization, for two $A$-modules $E$, $F$,

  𝕊(E ⊕ F) = 𝕊(E) ⊗ 𝕊(F)

up to canonical isomorphism. Furthermore, $\mathbb{S}(E)$ is a covariant functor in $E$ from $A$-modules to commutative $A$-algebras, and if $E=\underrightarrow{\lim }{E}_{\lambda }$, then $\mathbb{S}(E)=\underrightarrow{\lim }\mathbb{S}(E_{\lambda })$ up to canonical isomorphism. By abuse of notation, a product $\sigma (x_1)\sigma (x_2)\cdots \sigma (x_n)$ with $x_i\in E$ is often written $x_1{x}_2\cdots x_n$ when no confusion can arise. The algebra $\mathbb{S}(E)$ is graded, with ${\mathbb{S}}_n(E)$ the set of linear combinations of products of $n$ elements of $E$ ($n\ge 0$); $\mathbb{S}(A)$ is canonically isomorphic to the polynomial algebra A[T] in one indeterminate, and $\mathbb{S}(A^n)$ to $A[T_1,\cdots ,T_n]$.

(1.7.2)

Let $\varphi :A\to B$ be a ring homomorphism. If $F$ is a $B$-module, the canonical map $F\to \mathbb{S}(F)$ gives a canonical map $F_{[\varphi ]}\to \mathbb{S}(F{)}_{[\varphi ]}$, which factors as $F_{[\varphi ]}\to \mathbb{S}(F_{[\varphi ]})\to \mathbb{S}(F{)}_{[\varphi ]}$; the canonical homomorphism $\mathbb{S}(F_{[\varphi ]})\to \mathbb{S}(F{)}_{[\varphi ]}$ is surjective but not in general bijective. If $E$ is an $A$-module, every di-homomorphism $E\to F$ (that is, every $A$-homomorphism $E\to F_{[\varphi ]}$) yields canonically an $A$-homomorphism of algebras $\mathbb{S}(E)\to \mathbb{S}(F_{[\varphi ]})\to \mathbb{S}(F{)}_{[\varphi ]}$, i.e. a di-homomorphism of algebras $\mathbb{S}(E)\to \mathbb{S}(F)$.

With the same notation, for every $A$-module $E$, $\mathbb{S}(E{\otimes }_{A}B)$ is canonically isomorphic to $\mathbb{S}(E){\otimes }_{A}B$; this follows immediately from the universal property of $\mathbb{S}(E)$ (1.7.1).

(1.7.3)

Let $R$ be a multiplicative subset of $A$. Applying (1.7.2) with $B=R^{-1}A$ and recalling that $R^{-1}E=E{\otimes }_A{R}^{-1}A$, we get $\mathbb{S}(R^{-1}E)=R^{-1}\mathbb{S}(E)$ up to canonical isomorphism. If $R^{\prime }\supset R$ is a second multiplicative subset, the diagram

   R⁻¹E ─────→ R'⁻¹E
     │           │
     ↓           ↓
   𝕊(R⁻¹E) → 𝕊(R'⁻¹E)

commutes.

(1.7.4)

Now let $(S,\mathcal{A})$ be a ringed space and $\mathcal{E}$ an $\mathcal{A}$-module on $S$. Associating to every open $U\subset S$ the $\Gamma (U,\mathcal{A})$-module $\mathbb{S}(\Gamma (U,\mathcal{E}))$ defines (by the functoriality of $\mathbb{S}(E)$ (1.7.2)) a presheaf of algebras; we call the associated sheaf, denoted $\mathbb{S}(\mathcal{E})$ or ${\mathbb{S}}_{\mathcal{A}}(\mathcal{E})$, the symmetric $\mathcal{A}$-algebra on the $\mathcal{A}$-module $\mathcal{E}$. By (1.7.1) $\mathbb{S}(\mathcal{E})$ is the solution of a universal problem: every homomorphism of $\mathcal{A}$-modules $\mathcal{E}\to \mathcal{B}$ with $\mathcal{B}$ an $\mathcal{A}$-algebra factors uniquely as $\mathcal{E}\to \mathbb{S}(\mathcal{E})\to \mathcal{B}$, the second arrow being a homomorphism of $\mathcal{A}$-algebras. There is thus a bijective correspondence between homomorphisms $\mathcal{E}\to \mathcal{B}$ of $\mathcal{A}$-modules and homomorphisms $\mathbb{S}(\mathcal{E})\to \mathcal{B}$ of $\mathcal{A}$-algebras. In particular, every homomorphism $u:\mathcal{E}\to \mathcal{F}$ of $\mathcal{A}$-modules defines a homomorphism $\mathbb{S}(u):\mathbb{S}(\mathcal{E})\to \mathbb{S}(\mathcal{F})$ of $\mathcal{A}$-algebras, and $\mathbb{S}(\mathcal{E})$ is a covariant functor in $\mathcal{E}$.

By (1.7.2) and the commutation of $\mathbb{S}$ with direct limits, $(\mathbb{S}(\mathcal{E}){)}_s=\mathbb{S}({\mathcal{E}}_s)$ for every $s\in S$. For two $\mathcal{A}$-modules $\mathcal{E}$, $\mathcal{F}$, $\mathbb{S}(\mathcal{E}\oplus \mathcal{F})$ identifies canonically with $\mathbb{S}(\mathcal{E}){\otimes }_{\mathcal{A}}\mathbb{S}(\mathcal{F})$, as one sees on presheaves.

We also note that $\mathbb{S}(\mathcal{E})$ is a graded $\mathcal{A}$-algebra — the infinite direct sum of the ${\mathbb{S}}_n(\mathcal{E})$, where ${\mathbb{S}}_n(\mathcal{E})$ is the sheaf associated to the presheaf $U\mapsto {\mathbb{S}}_n(\Gamma (U,\mathcal{E}))$. In particular ${\mathbb{S}}_{\mathcal{A}}(\mathcal{A})$ identifies with $\mathcal{A}[T]=\mathcal{A}{\otimes }_{\mathbb{Z}}\mathbb{Z}[T]$ ($T$ an indeterminate, $\mathbb{Z}$ viewed as a constant sheaf).

(1.7.5)

Let $(T,\mathcal{B})$ be a second ringed space and $f:(S,\mathcal{A})\to (T,\mathcal{B})$ a morphism. For $\mathcal{F}$ a $\mathcal{B}$-module, $\mathbb{S}(f\ast (\mathcal{F}))$ identifies canonically with $f\ast (\mathbb{S}(\mathcal{F}))$: with $f=(\psi ,\theta )$ and by definition (0, 4.3.1),

  𝕊(f*(ℱ)) = 𝕊(ψ*(ℱ) ⊗_{ψ*(ℬ)} 𝒜) = 𝕊(ψ*(ℱ)) ⊗_{ψ*(ℬ)} 𝒜

by (1.7.2). For every open $U\subset S$ and every section $h$ of $\mathbb{S}(\psi \ast (\mathcal{F}))$ over $U$, $h$ agrees in a neighbourhood $V$ of every $s\in U$ with an element of $\mathbb{S}(\Gamma (V,\psi \ast (\mathcal{F})))$; unfolding the definition of $\psi \ast (\mathcal{F})$ (0, 3.7.1) and using that every element of $\mathbb{S}(E)$ is a linear combination of finitely many products of elements of $E$, one finds a neighbourhood $W$ of $\psi (s)$ in $T$, a section $h^{\prime }$ of $\mathbb{S}(\mathcal{F})$ over $W$, and a neighbourhood $V^{\prime }\subset V\cap {\psi }^{-1}(W)$ of $s$ such that $h$ agrees with $t\mapsto h^{\prime }(\psi (t))$ on $V^{\prime }$; whence the assertion.

Proposition.

Let $A$ be a ring, $S=\mathrm{Spec}(A)$ its prime spectrum, and $\mathcal{E}=\tilde{M}$ the ${\mathcal{O}}_S$-module associated to an $A$-module $M$. Then the ${\mathcal{O}}_S$-algebra $\mathbb{S}(\mathcal{E})$ is associated to the $A$-algebra $\mathbb{S}(M)$.

Proof. For every $f\in A$, $\mathbb{S}(M_f)=(\mathbb{S}(M){)}_f$ (1.7.3), so the proposition follows from (I, 1.3.4).

Corollary.

If $S$ is a prescheme and $\mathcal{E}$ a quasi-coherent ${\mathcal{O}}_S$-module, then the ${\mathcal{O}}_S$-algebra $\mathbb{S}(\mathcal{E})$ is quasi-coherent. If furthermore $\mathcal{E}$ is of finite type, then each ${\mathbb{S}}_n(\mathcal{E})$ is an ${\mathcal{O}}_S$-module of finite type.

Proof. The first assertion is immediate from (1.7.6) and (I, 1.4.1). The second follows because for an $A$-module $E$ of finite type, ${\mathbb{S}}_n(E)$ is also of finite type; apply (I, 1.3.13).

Definition.

Let $\mathcal{E}$ be a quasi-coherent ${\mathcal{O}}_S$-module. We call the vector bundle over $S$ defined by $\mathcal{E}$, denoted $\mathbb{V}(\mathcal{E})$, the spectrum (1.3.1) of the quasi-coherent ${\mathcal{O}}_S$-algebra $\mathbb{S}(\mathcal{E})$.

By (1.2.7), for every $S$-prescheme $X$ there is a canonical bijective correspondence between the $S$-morphisms $X\to \mathbb{V}(\mathcal{E})$ and the homomorphisms $\mathbb{S}(\mathcal{E})\to \mathcal{A}(X)$ of ${\mathcal{O}}_S$-algebras, hence also between these $S$-morphisms and the homomorphisms $\mathcal{E}\to \mathcal{A}(X)=f_{\ast }({\mathcal{O}}_X)$ of ${\mathcal{O}}_S$-modules (where $f$ is the structure morphism $X\to S$). In particular:

(1.7.9)

Take $X$ to be the subprescheme induced by $S$ on an open $U\subset S$. Then the $S$-morphisms $U\to \mathbb{V}(\mathcal{E})$ are just the $U$-sections (I, 2.5.5) of the $U$-prescheme induced by $\mathbb{V}(\mathcal{E})$ on $p^{-1}(U)$ (with $p:\mathbb{V}(\mathcal{E})\to S$ the structure morphism). By the above, these $U$-sections correspond bijectively to homomorphisms of ${\mathcal{O}}_S$-modules $\mathcal{E}\to j_{\ast }({\mathcal{O}}_S|U)$ (with $j:U\to S$ the canonical injection), or

equivalently (0, 4.4.3) to $({\mathcal{O}}_S|U)$-homomorphisms $j\ast (\mathcal{E})=\mathcal{E}|U\to {\mathcal{O}}_S|U$. It is immediate that restriction to an open $U^{\prime }\subset U$ of an $S$-morphism $U\to \mathbb{V}(\mathcal{E})$ corresponds to the restriction to $U^{\prime }$ of the corresponding homomorphism $\mathcal{E}|U\to {\mathcal{O}}_S|U$. We conclude that the sheaf of germs of $S$-sections of $\mathbb{V}(\mathcal{E})$ is canonically identified with the dual $\check{\mathcal{E}}$ of $\mathcal{E}$.

In particular, taking $X=U=S$, the zero homomorphism $\mathcal{E}\to {\mathcal{O}}_S$ corresponds to a canonical $S$-section of $\mathbb{V}(\mathcal{E})$, called the zero $S$-section (cf. (8.3.3)).

(1.7.10)

Now take $X$ to be the spectrum $\xi$ of a field $K$; the structure morphism $f:X\to S$ corresponds to a monomorphism $\kappa (s)\to K$ with $s=f(\xi )$ (I, 2.4.6). The $S$-morphisms $\xi \to \mathbb{V}(\mathcal{E})$ are then just the geometric points of $\mathbb{V}(\mathcal{E})$ with values in the extension $K$ of $\kappa (s)$ (I, 3.4.5), points localized over $p^{-1}(s)$. The set of these points — which we call the rational geometric fibre over $K$ of $\mathbb{V}(\mathcal{E})$ over $s$ — identifies, by (1.7.8), with ${\mathrm{Hom}}_{{\mathcal{O}}_S}(\mathcal{E},f_{\ast }({\mathcal{O}}_X))$, or equivalently (0, 4.4.3) with ${\mathrm{Hom}}_{{\mathcal{O}}_X}(f\ast (\mathcal{E}),{\mathcal{O}}_X=K)$. By definition (0, 4.3.1) $f\ast (\mathcal{E})={\mathcal{E}}_s{\otimes }_{{\mathcal{O}}_s}K={\mathcal{E}}^s{\otimes }_{\kappa (s)}K$ with ${\mathcal{E}}^s={\mathcal{E}}_s/{\mathfrak{m}}_s{\mathcal{E}}_s$; so the rational geometric fibre of $\mathbb{V}(\mathcal{E})$ over $s$ with values in $K$ identifies with the dual of the $K$-vector space ${\mathcal{E}}^s{\otimes }_{\kappa (s)}K$. If ${\mathcal{E}}^s$ or $K$ is finite-dimensional over $\kappa (s)$, this dual identifies further with $\check{{\mathcal{E}}^s}{\otimes }_{\kappa (s)}K$, where $\check{{\mathcal{E}}^s}$ denotes the dual of the $\kappa (s)$-vector space ${\mathcal{E}}^s$.

Proposition.

1. $\mathbb{V}(\mathcal{E})$ is a contravariant functor in $\mathcal{E}$ from quasi-coherent ${\mathcal{O}}_S$-modules to affine $S$-schemes.
2. If $\mathcal{E}$ is an ${\mathcal{O}}_S$-module of finite type, then $\mathbb{V}(\mathcal{E})$ is of finite type over $S$.
3. If $\mathcal{E}$ and $\mathcal{F}$ are two quasi-coherent ${\mathcal{O}}_S$-modules, then $\mathbb{V}(\mathcal{E}\oplus \mathcal{F})$ identifies canonically with $\mathbb{V}(\mathcal{E}){\times }_S\mathbb{V}(\mathcal{F})$.
4. For a morphism $g:S^{\prime }\to S$ and any quasi-coherent ${\mathcal{O}}_S$-module $\mathcal{E}$, $\mathbb{V}(g\ast (\mathcal{E}))$ identifies canonically with $\mathbb{V}(\mathcal{E}{)}_{(S^{\prime })}=\mathbb{V}(\mathcal{E}){\times }_S{S}^{\prime }$.
5. A surjective homomorphism $\mathcal{E}\to \mathcal{F}$ of quasi-coherent ${\mathcal{O}}_S$-modules corresponds to a closed immersion $\mathbb{V}(\mathcal{F})\to \mathbb{V}(\mathcal{E})$.

Proof. (i) is immediate from (1.2.7), given that every homomorphism $\mathcal{E}\to \mathcal{F}$ of ${\mathcal{O}}_S$-modules canonically defines a homomorphism $\mathbb{S}(\mathcal{E})\to \mathbb{S}(\mathcal{F})$ of ${\mathcal{O}}_S$-algebras. (ii) follows immediately from (I, 6.3.1) and the fact that for an $A$-module $E$ of finite type, $\mathbb{S}(E)$ is an $A$-algebra of finite type. For (iii), start from the canonical isomorphism $\mathbb{S}(\mathcal{E}\oplus \mathcal{F})\stackrel{\sim }{\to }\mathbb{S}(\mathcal{E}){\otimes }_{{\mathcal{O}}_S}\mathbb{S}(\mathcal{F})$ (1.7.4) and apply (1.4.6). Similarly, for (iv), start from $\mathbb{S}(g\ast (\mathcal{E}))\stackrel{\sim }{\to }g\ast (\mathbb{S}(\mathcal{E}))$ (1.7.5) and apply (1.5.2). Finally, for (v), surjectivity of $\mathcal{E}\to \mathcal{F}$ implies surjectivity of the corresponding homomorphism $\mathbb{S}(\mathcal{E})\to \mathbb{S}(\mathcal{F})$ of ${\mathcal{O}}_S$-algebras, and the conclusion follows from (1.4.10).

(1.7.12)

Taking $\mathcal{E}={\mathcal{O}}_S$ in particular, the prescheme $\mathbb{V}({\mathcal{O}}_S)$ is the affine $S$-scheme which is the spectrum of the ${\mathcal{O}}_S$-algebra $\mathbb{S}({\mathcal{O}}_S)$, and the latter identifies with the ${\mathcal{O}}_S$-algebra ${\mathcal{O}}_S[T]={\mathcal{O}}_S{\otimes }_{\mathbb{Z}}\mathbb{Z}[T]$

($T$ an indeterminate). This is evident when $S=\mathrm{Spec}(\mathbb{Z})$, by (1.7.6), and the general case follows by considering the structure morphism $S\to \mathrm{Spec}(\mathbb{Z})$ and using (1.7.11)(iv). We therefore write $\mathbb{V}({\mathcal{O}}_S)=S[T]$, and we have

  S[T] = S ⊗_ℤ ℤ[T].                                                      (1.7.12.1)

The identification of the sheaf of germs of $S$-sections of S[T] with ${\mathcal{O}}_S$, already seen in (I, 3.3.15), reappears here in a more general context, as a special case of (1.7.9).

(1.7.13)

For every $S$-prescheme $X$, we have seen (1.7.8) that ${\mathrm{Hom}}_S(X,S[T])$ is canonically identified with ${\mathrm{Hom}}_{{\mathcal{O}}_S}({\mathcal{O}}_S,\mathcal{A}(X))$, which is canonically isomorphic to $\Gamma (S,\mathcal{A}(X))$ and so carries a ring structure. To every $S$-morphism $h:X\to Y$ corresponds a morphism $\Gamma (\mathcal{A}(h)):\Gamma (S,\mathcal{A}(Y))\to \Gamma (S,\mathcal{A}(X))$ for these ring structures (1.1.2). Equipped with this ring structure, ${\mathrm{Hom}}_S(X,S[T])$ becomes a contravariant functor in $X$ from the category of $S$-preschemes to rings. On the other hand, ${\mathrm{Hom}}_S(X,\mathbb{V}(\mathcal{E}))$ identifies likewise with ${\mathrm{Hom}}_{{\mathcal{O}}_S}(\mathcal{E},\mathcal{A}(X))$ (with $\mathcal{A}(X)$ viewed as an ${\mathcal{O}}_S$-module); it is therefore canonically endowed with a module structure over the ring ${\mathrm{Hom}}_S(X,S[T])$, and the pair

  (Hom_S(X, S[T]), Hom_S(X, 𝕍(ℰ)))

is a contravariant functor in $X$ with values in the category whose objects are pairs $(A,M)$ with $A$ a ring and $M$ an $A$-module, the morphisms being di-homomorphisms.

We interpret this by saying that S[T] is an $S$-scheme of rings and that $\mathbb{V}(\mathcal{E})$ is an $S$-scheme of modules over the $S$-scheme of rings S[T] (cf. Chapter 0, §8).

(1.7.14)

We shall see that the $S$-scheme-of-modules structure on $\mathbb{V}(\mathcal{E})$ reconstructs $\mathcal{E}$ up to unique isomorphism: we show that $\mathcal{E}$ is canonically isomorphic to an ${\mathcal{O}}_S$-submodule of $\mathbb{S}(\mathcal{E})=\mathcal{A}(\mathbb{V}(\mathcal{E}))$ defined by means of that structure. Indeed (1.7.4), the set ${\mathrm{Hom}}_{{\mathcal{O}}_S}(\mathbb{S}(\mathcal{E}),\mathcal{A}(X))$ of homomorphisms of ${\mathcal{O}}_S$-algebras identifies canonically with ${\mathrm{Hom}}_{{\mathcal{O}}_S}(\mathcal{E},\mathcal{A}(X))$, the set of homomorphisms of ${\mathcal{O}}_S$-modules: if $h$, $h^{\prime }$ are two elements of the latter set, $s_i$ ($1\le i\le n$) sections of $\mathcal{E}$ over an open $U\subset S$, and $t$ a section of $\mathcal{A}(X)$ over $U$, then by definition

  (h + h')(s_1 s_2 ⋯ s_n) = ∏_{i=1}^n (h(s_i) + h'(s_i))

and

  (t · h)(s_1 s_2 ⋯ s_n) = tⁿ ∏_{i=1}^n h(s_i).

Given this, if $z$ is a section of $\mathbb{S}(\mathcal{E})$ over $U$, then $h\mapsto h(z)$ is a map from ${\mathrm{Hom}}_S(X,\mathbb{V}(\mathcal{E}))={\mathrm{Hom}}_{{\mathcal{O}}_S}(\mathbb{S}(\mathcal{E}),\mathcal{A}(X))$ to $\Gamma (U,\mathcal{A}(X))$. We will

show that $\mathcal{E}$ is identified with the ${\mathcal{O}}_S$-submodule of $\mathbb{S}(\mathcal{E})$ such that, for every open $U\subset S$, every section $z$ of this submodule over $U$, and every $S$-prescheme $X$, the map $h\mapsto h(z)$ from ${\mathrm{Hom}}_{{\mathcal{O}}_S|U}(\mathbb{S}(\mathcal{E})|U,\mathcal{A}(X)|U)$ to $\Gamma (U,\mathcal{A}(X))$ is a homomorphism of $\Gamma (U,\mathcal{A}(X))$-modules.

It is immediate that $\mathcal{E}$ has this property; for the converse, we reduce to proving that when $S=\mathrm{Spec}(A)$ and $\mathcal{E}=\tilde{M}$, a section $z$ of $\mathbb{S}(\mathcal{E})$ over $S$ (for $U=S$) satisfying the stated property must be a section of $\mathcal{E}$. Write $z={\sum }_{n=0}^{\infty }{z}_n$ with $z_n\in {\mathbb{S}}_n(M)$; we must show $z_n=0$ for $n\ne 1$. Set $B=\mathbb{S}(M)$ and take $X=\mathrm{Spec}(B[T])$ for an indeterminate $T$. Then ${\mathrm{Hom}}_{{\mathcal{O}}_S}(\mathbb{S}(\mathcal{E}),\mathcal{A}(X))$ identifies with the set of ring homomorphisms $h:B\to B[T]$ (I, 1.3.13), and by the calculation above $(T\cdot h)(z)={\sum }_{n=0}^{\infty }{T}^{n}h(z_n)$; the hypothesis on $z$ gives ${\sum }_{n=0}^{\infty }{T}^{n}h(z_n)=T\cdot {\sum }_{n=0}^{\infty }h(z_n)$ for every $h$. Taking $h$ to be the canonical injection yields ${\sum }_{n=0}^{\infty }{T}^n{z}_n=T\cdot {\sum }_{n=0}^{\infty }{z}_n$, which forces $z_n=0$ for $n\ne 1$.

Proposition.

Let $Y$ be a prescheme whose underlying space is Noetherian, or a quasi-compact scheme. Every affine $Y$-scheme $X$ of finite type over $Y$ is $Y$-isomorphic to a closed $Y$-subscheme of a $Y$-scheme of the form $\mathbb{V}(\mathcal{E})$, where $\mathcal{E}$ is a quasi-coherent ${\mathcal{O}}_Y$-module of finite type.

Proof. The quasi-coherent ${\mathcal{O}}_Y$-algebra $\mathcal{A}(X)$ is of finite type (1.3.7). The hypotheses imply $\mathcal{A}(X)$ is generated by a quasi-coherent ${\mathcal{O}}_Y$-submodule $\mathcal{E}$ of finite type (I, 9.6.5); by definition, the canonical homomorphism $\mathbb{S}(\mathcal{E})\to \mathcal{A}(X)$ extending the injection $\mathcal{E}\to \mathcal{A}(X)$ is surjective. The conclusion follows from (1.4.10).