Ch.4 §1. Relative finiteness conditions - Éléments de Géométrie Algébrique

§1. Relative finiteness conditions. Constructible sets in preschemes

In this section we resume, in completed form, the exposition of the "finiteness conditions" for a morphism of preschemes $f : X \to Y$ given in (I, 6.3 and 6.6). There are essentially two notions of "finiteness" of global nature on $X$ , that of quasi-compact morphism (defined in (I, 6.6.1)) and that of quasi-separated morphism; there are on the other hand two notions of "finiteness" of local nature on $X$ , that of morphism locally of finite type (defined in (I, 6.6.2)) and that of morphism locally of finite presentation. Combining these local notions with the preceding global notions, one obtains the notions of morphism of finite type (defined in (I, 6.3.1)) and of morphism of finite presentation. For the reader's convenience we shall give again, in this section, the principal properties stated in (I, 6.3 and 6.6), of course referring back to those numbers of Chapter I for their proofs.

In nos. (1.8) and (1.9) we complete, in the framework of preschemes and making use of the preceding finiteness notions, the results on constructible sets given in (0_III, §9).

1.1. Quasi-compact morphisms

Definition (1.1.1).

We say that a morphism of preschemes $f : X \to Y$ is quasi-compact if the continuous map $f$ from the topological space $X$ to the topological space $Y$ is quasi-compact $(0_{I}, 9.1.1)$ , in other words if the inverse image $f^{- 1} (U)$ of every quasi-compact open set $U$ of $Y$ is quasi-compact (cf. (I, 6.6.1)).

If $B$ is a base of the topology of $Y$ formed of affine open sets, then for $f$ to be quasi-compact it is necessary and sufficient that for every $V \in B$ , $f^{- 1} (V)$ be a finite union of affine open sets. For example, if $Y$ is affine and $X$ is quasi-compact, every morphism $f : X \to Y$ is quasi-compact (I, 6.6.1).

If $f : X \to Y$ is a quasi-compact morphism, it is clear that for every open set $V$ of $Y$ the restriction of $f$ to $f^{- 1} (V)$ is a quasi-compact morphism $f^{- 1} (V) \to V$ . Conversely, if $(U_{α})$ is an open cover of $Y$ and $f : X \to Y$ is a morphism such that the restrictions $f^{- 1} (U_{α}) \to U_{α}$ are quasi-compact, then $f$ is quasi-compact. Consequently, if $f : X \to Y$ is an $S$ -morphism of $S$ -preschemes and there exists an open cover $(S_{λ})$ of $S$ such that the restrictions $g^{- 1} (S_{λ}) \to h^{- 1} (S_{λ})$ of $f$ (where $g$ and $h$ are the structure morphisms) are quasi-compact, then $f$ is quasi-compact.

Proposition (1.1.2).

(i) An immersion $X \to Y$ is quasi-compact if it is closed, or if the space underlying $Y$ is locally Noetherian, or if the space underlying $X$ is Noetherian.

(ii) The composite of two quasi-compact morphisms is quasi-compact.

(iii) If $f : X \to Y$ is a quasi-compact $S$ -morphism, then so is $f_{(S^{'})} : X_{(S^{'})} \to Y_{(S^{'})}$ for every extension $g : S^{'} \to S$ of the base prescheme.

(iv) If $f : X \to X^{'}$ and $g : Y \to Y^{'}$ are two quasi-compact $S$ -morphisms,

  f ×_S g : X ×_S Y → X' ×_S Y'

is quasi-compact.

(v) If the composite $g \circ f$ of two morphisms $f : X \to Y$ , $g : Y \to Z$ is quasi-compact, and if $g$ is separated, or the space underlying $X$ locally Noetherian, then $f$ is quasi-compact.

(vi) For a morphism $f$ to be quasi-compact, it is necessary and sufficient that $f_{re d}$ be so.

For the proof, see (I, 6.6.4). We note that assertion (vi) is also a consequence of the more general proposition:

Proposition (1.1.3).

Let $f : X \to Y$ , $g : Y \to Z$ be two morphisms. If $g \circ f$ is quasi-compact and $f$ is surjective, then $g$ is quasi-compact.

Indeed, if $V$ is a quasi-compact open set in $Z$ , $f^{- 1} (g^{- 1} (V))$ is quasi-compact by hypothesis, and one has $g^{- 1} (V) = f (f^{- 1} (g^{- 1} (V)))$ , since $f$ is surjective; hence $g^{- 1} (V)$ is quasi-compact.

Corollary (1.1.4).

Let $f : X \to Y$ , $g : Y^{'} \to Y$ be two morphisms, $X^{'} = X \times_{Y} Y^{'}$ , $f^{'} = f_{(Y^{'})} : X^{'} \to Y^{'}$ ; suppose $g$ is quasi-compact and surjective (resp. surjective). Then, for $f$ to be quasi-compact (resp. dominant), it suffices that $f^{'}$ be so.

If $g^{'} : X^{'} \to X$ is the canonical projection, $g^{'}$ is a surjective morphism (I, 3.5.2), and one has $f \circ g^{'} = g \circ f^{'}$ ; if $f^{'}$ is quasi-compact (resp. dominant), so is $g \circ f^{'}$ , since $g$ is quasi-compact (1.1.2) (resp. surjective); since $f \circ g^{'}$ is quasi-compact (resp. dominant) and $g^{'}$ is surjective, one deduces from (1.1.3) that $f$ is quasi-compact (resp. dominant).

To abbreviate, we shall call a maximal point of a prescheme $X$ any generic point of an irreducible component of $X$ ; if $X = Spec (A)$ is affine, the maximal points of $X$ are the minimal prime ideals of $A$ .

Proposition (1.1.5).

Let $f : X \to Y$ be a quasi-compact morphism. The following conditions are equivalent:

a) $f$ is dominant;

b) for every maximal point $y$ of $Y$ , one has $f^{- 1} (y) \neq = \emptyset$ ;

c) for every maximal point $y$ of $Y$ , $f^{- 1} (y)$ contains a maximal point of $X$ .

The equivalence of a) and c) was proved in (I, 6.6.5). It is clear that c) implies b); on the other hand, if $X^{'}$ is an irreducible component of $X$ meeting $f^{- 1} (y)$ , then $f (X^{'})$ is irreducible and contains $y$ , so it is contained in the irreducible component $Y^{'} = \overline{y}$ of $Y$ $(0_{I}, 2.1.6)$ ; if $x$ is the generic point of $X^{'}$ , one therefore has necessarily $f (x) = y$ $(0_{I}, 2.1.5)$ ; hence b) implies c).

Proposition (1.1.6).

Let $f^{'} : X^{'} \to Y$ , $f^{''} : X^{''} \to Y$ be two morphisms, $X$ the sum prescheme $X^{'} ⨿ X^{''}$ , $f : X \to Y$ the morphism equal to $f^{'}$ on $X^{'}$ and to f'' on X''. For $f$ to be quasi-compact it is necessary and sufficient that $f^{'}$ and f'' be so.

This results immediately from the definition.

1.2. Quasi-separated morphisms

Definition (1.2.1).

Let $X$ , $Y$ be two preschemes. We say that a morphism $f : X \to Y$ is quasi-separated (or that $X$ is quasi-separated over $Y$ ) if the diagonal morphism $Δ_{f} = (1_{X}, 1_{X})_{Y} : X \to X \times_{Y} X$ is quasi-compact. We say that a prescheme $X$ is quasi-separated if it is quasi-separated over $Spec (Z)$ .

By definition, every separated morphism is quasi-separated, $Δ_{f}$ being a closed immersion (1.1.2, (i)); a scheme $X$ is a quasi-separated prescheme, being separated over $Spec (Z)$ (I, 5.4.1).

Proposition (1.2.2).

(i) Every monomorphism of preschemes $f : X \to Y$ (in particular, every immersion) is quasi-separated.

(ii) If $f : X \to Y$ and $g : Y \to Z$ are two quasi-separated morphisms, then $g \circ f : X \to Z$ is quasi-separated.

(iii) Let $X$ , $Y$ be two $S$ -preschemes, $f : X \to Y$ a quasi-separated $S$ -morphism. Then, for every base change $g : S^{'} \to S$ , the morphism $f_{(S^{'})} : X_{(S^{'})} \to Y_{(S^{'})}$ is quasi-separated.

(iv) If $f : X \to Y$ , $f^{'} : X^{'} \to Y^{'}$ are two quasi-separated $S$ -morphisms, then

  f ×_S f' : X ×_S X' → Y ×_S Y'

is quasi-separated.

(v) If $f : X \to Y$ , $g : Y \to Z$ are two morphisms such that $g \circ f$ is quasi-separated, then $f$ is quasi-separated.

(vi) If $f : X \to Y$ is a quasi-separated morphism, then f_red : X_red → Y_red is quasi-separated.

By virtue of (I, 5.5.12), it suffices to prove (i), (ii), and (iii).

Assertion (i) is trivial, every monomorphism being separated (I, 5.5.1). To prove (iii), one reduces immediately to the case where $Y = S$ (I, 3.3.11), and the assertion results from the fact that $Δ_{f^{'}} = (Δ_{f})_{(S^{'})}$ (I, 5.3.4) and from (1.1.2, (iii)). To prove (ii), consider the projections $p$ and $q$ from $X \times_{Y} X$ onto $X$ ; if $i = (p, q)_{Z}$ , one knows that the diagram

  X ×_Y X  ──i──→  X ×_Z X
    │                │
    π│              │f ×_Z f
    ↓                ↓
    Y    ──Δ_g──→  Y ×_Z Y

(where $π$ is the structure morphism) is commutative and identifies $X \times_{Y} X$ with the product of the $(Y \times_{Z} Y)$ -preschemes $Y$ and $X \times_{Z} X$ (I, 5.3.5). If $g$ is quasi-separated, $Δ_{g}$ is quasi-compact,

so the same is true of $i$ (1.1.2, (iii)); if in addition $f$ is quasi-separated, $Δ_{f}$ is quasi-compact, hence so is $i \circ Δ_{f}$ (1.1.2, (ii)), which is equal to $Δ_{g \circ f}$ .

Corollary (1.2.3).

(i) Let $X$ be a quasi-separated prescheme. Then every morphism $f : X \to Y$ is quasi-separated.

(ii) Let $Y$ be a quasi-separated prescheme. For a morphism $f : X \to Y$ to be quasi-separated, it is necessary and sufficient that the prescheme $X$ be quasi-separated.

This results at once from (1.2.2, (ii) and (v)) applied to $X$ , $Y$ , and $Spec (Z)$ .

Proposition (1.2.4).

Let $f : X \to Y$ be a morphism, $g : Y \to Z$ a quasi-separated morphism. If $g \circ f$ is quasi-compact, then so is $f$ .

One knows that $f$ is the composite morphism $X Γ_{f} X \times_{Z} Y p r_{2} Y$ (I, 5.3.13); on the other hand, $p r_{2}$ identifies with $(g \circ f) \times_{Z} 1_{Y}$ (I, 3.3.4), and if $g \circ f$ is quasi-compact, then so is $p r_{2}$ (1.1.2, (iv)). Finally, one has the commutative diagram

  X        ──Γ_f──→  X ×_Z Y                                    (1.2.4.1)
  │                    │
  f│                  │f × 1_Y
  ↓                    ↓
  Y        ──Δ_g──→  Y ×_Z Y

which identifies $X$ with the product of the $(Y \times_{Z} Y)$ -preschemes $Y$ and $X \times_{Z} Y$ (I, 5.3.7). Since by hypothesis $Δ_{g}$ is a quasi-compact morphism, so is $Γ_{f}$ (1.1.2, (iii)); the conclusion therefore follows from (1.1.2, (ii)).

Proposition (1.2.5).

Let $f : X \to Y$ , $g : Y^{'} \to Y$ be two morphisms, $X^{'} = X \times_{Y} Y^{'}$ , $f^{'} = f_{(Y^{'})} : X^{'} \to Y^{'}$ . If $g$ is quasi-compact and surjective and $f^{'}$ is quasi-separated, then $f$ is quasi-separated.

Indeed, one has $(X^{'} \times_{Y^{'}} X^{'}) = (X \times_{Y} X)_{(Y^{'})}$ and $Δ_{f^{'}} = (Δ_{f})_{(Y^{'})}$ (I, 5.3.4); as the projection $X^{'} \times_{Y^{'}} X^{'} \to X \times_{Y} X$ is quasi-compact (1.1.2, (iii)) and surjective (I, 3.5.2), one may, by virtue of (I, 3.3.11), apply (1.1.4), which shows that since $Δ_{f^{'}}$ is a quasi-compact morphism, so is $Δ_{f}$ .

Proposition (1.2.6).

Let $f : X \to Y$ be a morphism, $(U_{α})$ a cover of $Y$ by open sets such that the induced preschemes $U_{α}$ are quasi-separated. For $f$ to be quasi-separated it is necessary and sufficient that each of the preschemes induced by $X$ on the open sets $f^{- 1} (U_{α})$ be quasi-separated.

The inverse image in $X \times_{Y} X$ of $U_{α}$ is $X_{α} \times_{U_{α}} X_{α}$ , where $X_{α} = f^{- 1} (U_{α})$ , and the restriction $X_{α} \to X_{α} \times_{U_{α}} X_{α}$ of $Δ_{f}$ is none other than $Δ_{f_{α}}$ , denoting by $f_{α}$ the restriction $X_{α} \to U_{α}$ of $f$ . By virtue of definition (1.2.1) and of the local character of the notion of quasi-compact morphism (1.1.1), for $f$ to be quasi-separated it is necessary and sufficient that each of the morphisms $f_{α}$ be so. But since by hypothesis the morphism $U_{α} \to Spec (Z)$ is quasi-separated, to say that $f_{α}$ is quasi-separated is the same as to say that the composite $X_{α} f_{α} U_{α} \to Spec (Z)$ is so (1.2.2, (ii) and (v)), whence the proposition.

To check that a morphism is quasi-separated, one is therefore reduced to giving criteria for a prescheme to be quasi-separated:

Proposition (1.2.7).

Let $X$ be a prescheme, $(U_{α})$ a cover of $X$ formed of quasi-compact open sets. The following properties are equivalent:

a) $X$ is quasi-separated.

b) For every quasi-compact open set $U$ of $X$ , the canonical immersion $U \to X$ is quasi-compact (in other words, $U$ is retrocompact in $X$ ).

b') The intersection of two quasi-compact open sets of $X$ is quasi-compact.

c) For every pair of indices $(α, β)$ , the intersection $U_{α} \cap U_{β}$ is quasi-compact.

Properties b) and b') are trivially equivalent by definition of a quasi-compact morphism. Since a quasi-compact open set is a finite union of affine open sets, for two quasi-compact open sets $U$ , $V$ of $X$ , $U \times_{Z} V$ is a quasi-compact open set of $X \times_{Z} X$ (I, 3.2.7), whose inverse image under $Δ_{X}$ is $U \cap V$ ; hence a) implies b'). It is trivial that b') implies c); finally, if c) holds, the $U_{α} \times_{Z} U_{β}$ form a cover of $X \times_{Z} X$ by quasi-compact open sets, and one knows that for the morphism $Δ_{X} : X \to X \times_{Z} X$ to be quasi-compact it suffices that the inverse images $U_{α} \cap U_{β}$ of the $U_{α} \times_{Z} U_{β}$ under $Δ_{X}$ be quasi-compact (1.1.1); hence c) implies a).

Corollary (1.2.8).

Every prescheme $X$ whose underlying space is locally Noetherian (for example, a locally Noetherian prescheme) is quasi-separated; every morphism $f : X \to Y$ is then quasi-separated.

Proposition (1.2.9).

Let $X^{'}$ , X'' be two preschemes, $X = X^{'} ⨿ X^{''}$ their sum, $f^{'} : X^{'} \to Y$ , $f^{''} : X^{''} \to Y$ two morphisms, $f : X \to Y$ the morphism that coincides with $f^{'}$ on $X^{'}$ and with f'' on X''. For $f$ to be quasi-separated it is necessary and sufficient that $f^{'}$ and f'' be so.

Indeed, $X \times_{Y} X$ is the sum of the four preschemes $X^{'} \times_{Y} X^{'}$ , $X^{'} \times_{Y} X^{''}$ , $X^{''} \times_{Y} X^{'}$ , and $X^{''} \times_{Y} X^{''}$ , and $Δ_{f}$ is the morphism that coincides with $Δ_{f^{'}}$ on $X^{'}$ and with $Δ_{f^{''}}$ on X''; the proposition therefore results at once from the definitions.

1.3. Morphisms locally of finite type

(1.3.1)

Recall that if $A$ is a ring, an (commutative) $A$ -algebra $B$ is said to be of finite type if it is generated by a finite number of elements of $B$ , or — what amounts to the same — if it is isomorphic to a quotient of a polynomial algebra $A [T_{1}, \dots, T_{n}]$ by an ideal of that algebra. It is clear that for every (commutative) $A$ -algebra $A^{'}$ , $B \otimes_{A} A^{'}$ is then an $A^{'}$ -algebra of finite type. If $B$ is an $A$ -algebra of finite type and $C$ a $B$ -algebra of finite type, then $C$ is an $A$ -algebra of finite type; for if $C$ is a quotient of $B [T_{1}, \dots, T_{n}]$ and $B$ a quotient of $A [S_{1}, \dots, S_{m}]$ , then $B [T_{1}, \dots, T_{n}] = B \otimes_{A} A [T_{1}, \dots, T_{n}]$ is a quotient of $A [S_{1}, \dots, S_{m}, T_{1}, \dots, T_{n}]$ , hence so is $C$ .

Definition (1.3.2).

Let $f : X \to Y$ be a morphism of preschemes, $x$ a point of $X$ , $y = f (x)$ . We say that $f$ is of finite type at $x$ if there exist an affine open neighbourhood $V$ of $y$ and an affine open neighbourhood $U$ of $x$ such that $f (U) \subset V$ and the ring $A (U)$ is an $A (V)$ -algebra of finite type. We say that $f$ is locally of finite type if it is of finite type at every point of $X$ .

Proposition (1.3.3).

If $Y$ is a locally Noetherian prescheme and $f : X \to Y$ a morphism locally of finite type, then $X$ is locally Noetherian.

For the proof, see (I, 6.3.7).

Proposition (1.3.4).

(i) Every local immersion is locally of finite type.

(ii) If two morphisms $f : X \to Y$ , $g : Y \to Z$ are locally of finite type, then so is $g \circ f$ .

(iii) If $f : X \to Y$ is an $S$ -morphism locally of finite type, then $f_{(S^{'})} : X_{(S^{'})} \to Y_{(S^{'})}$ is locally of finite type for every extension $S^{'} \to S$ of the base prescheme.

(iv) If $f : X \to X^{'}$ and $g : Y \to Y^{'}$ are two $S$ -morphisms locally of finite type, then $f \times_{S} g$ is locally of finite type.

(v) If the composite $g \circ f$ of two morphisms is locally of finite type, then $f$ is locally of finite type.

(vi) If a morphism $f$ is locally of finite type, then so is $f_{re d}$ .

For the proof, see (I, 6.6.6).

Corollary (1.3.5).

Let $f : X \to Y$ be a morphism locally of finite type. For every morphism $Y^{'} \to Y$ such that $Y^{'}$ is locally Noetherian, $X \times_{Y} Y^{'}$ is locally Noetherian.

Indeed, $f_{(Y^{'})} : X \times_{Y} Y^{'} \to Y^{'}$ is locally of finite type by (1.3.4, (iii)), and it suffices to apply (1.3.3).

Proposition (1.3.6).

Let $ϕ : A \to B$ be a ring homomorphism. For the corresponding morphism $f : Spec (B) \to Spec (A)$ to be locally of finite type, it is necessary and sufficient that $B$ be an $A$ -algebra of finite type.

For the proof, see (I, 6.3.3), taking into account that $f$ is quasi-compact and that (I, 6.6.3) holds.

Proposition (1.3.7).

For a morphism $f : X \to Y$ locally of finite type to be surjective, it is necessary and sufficient that for every algebraically closed field $Ω$ the map $X (Ω) \to Y (Ω)$ corresponding to $f$ (I, 3.4.1) be surjective.

For the proof, see (I, 6.3.10).

(1.3.8)

Let $A$ be a ring. We say that an $A$ -algebra $B$ is essentially of finite type if $B$ is $A$ -isomorphic to an $A$ -algebra of the form $S^{- 1} C$ , where $C$ is an $A$ -algebra of finite type and $S$ a multiplicative subset of $C$ .

Proposition (1.3.9).

(i) If $B$ is an $A$ -algebra essentially of finite type and $C$ a $B$ -algebra essentially of finite type, then $C$ is an $A$ -algebra essentially of finite type.

(ii) Let $B$ be an $A$ -algebra essentially of finite type, $A^{'}$ an $A$ -algebra; then $B^{'} = B \otimes_{A} A^{'}$ is an $A^{'}$ -algebra essentially of finite type.

(i) Let $B = S^{' - 1} B^{'}$ and $C = T^{' - 1} C^{'}$ , where $B^{'}$ (resp. $C^{'}$ ) is an $A$ -algebra (resp. a $B$ -algebra) of finite type and $S^{'}$ (resp. $T^{'}$ ) a multiplicative subset of $B^{'}$ (resp. $C^{'}$ ); then $C^{'}$ is of the form $S^{' - 1} C^{''}$ , where C'' is a $B^{'}$ -algebra of finite type, so $C$ is also a ring of fractions of C''; since C'' is an $A$ -algebra of finite type, this proves our assertion.

(ii) If $B$ is a ring of fractions of an $A$ -algebra of finite type $C$ , then $B^{'}$ is a ring of fractions of the $A^{'}$ -algebra of finite type $C \otimes_{A} A^{'}$ , whence (ii).

(1.3.10)

If $B$ is a local $A$ -algebra essentially of finite type, then $B$ is of the form $C_{q}$ , where $C$ is an $A$ -algebra of finite type and $q$ a prime ideal of $C$ (Bourbaki, Alg. comm., chap. II, §5, prop. 11). Let $p$ be the inverse image in $A$ of the ideal $q$ ; setting $S = A - p$ , $C_{q}$ is also a local ring at a prime ideal of $S^{- 1} C$ ; since $S^{- 1} C$ is an algebra of finite type over $A_{p} = S^{- 1} A$ , one sees that $B$ is also an $A_{p}$ -algebra essentially of finite type, and in addition the homomorphism $A_{p} \to B$ is local.

Proposition (1.3.11).

If $B$ is a local $A$ -algebra essentially of finite type, then $B$ is $A$ -isomorphic to a quotient $A$ -algebra of an $A$ -algebra of the form $C_{q}$ , where $C$ is a polynomial algebra $A [T_{1}, \dots, T_{n}]$ and $q$ a prime ideal of $C$ .

Indeed, by definition, $B$ is isomorphic to $C_{q^{'}}^{'}$ , where $C^{'}$ is an $A$ -algebra of finite type and $q^{'}$ a prime ideal of $C^{'}$ ; but $C^{'} = C / b$ , where $C = A [T_{1}, \dots, T_{n}]$ and $b$ is an ideal of $C$ ; so $q^{'} = q / b$ , where $q$ is a prime ideal of $C$ , and $C_{q^{'}}^{'}$ is isomorphic to $C_{q} / b C_{q}$ .

1.4. Morphisms locally of finite presentation

(1.4.1)

Given a ring $A$ , we shall say that an (commutative) $A$ -algebra $B$ is of finite presentation if it is isomorphic to the quotient $A [T_{1}, \dots, T_{n}] / a$ of a polynomial algebra over $A$ by an ideal of finite type $a$ of $A [T_{1}, \dots, T_{n}]$ . For every (commutative) $A$ -algebra $A^{'}$ , $B \otimes_{A} A^{'}$ is then an $A^{'}$ -algebra of finite presentation, being isomorphic to the quotient of $A^{'} [T_{1}, \dots, T_{n}]$ by the canonical image in this ring of $a \otimes_{A} A^{'}$ , which is manifestly an $A^{'} [T_{1}, \dots, T_{n}]$ -module of finite type. If $B$ is an $A$ -algebra of finite presentation and $C$ a $B$ -algebra of finite presentation, then $C$ is an $A$ -algebra of finite presentation. Indeed, let $B = A [S_{1}, \dots, S_{m}] / a$ and $C = B [T_{1}, \dots, T_{n}] / b$ , where $a$ (resp. $b$ ) is an ideal of finite type of $A [S_{1}, \dots, S_{m}]$ (resp. of $B [T_{1}, \dots, T_{n}]$ ); the ring $B [T_{1}, \dots, T_{n}]$ is isomorphic to $A [S_{1}, \dots, S_{m}, T_{1}, \dots, T_{n}] / a^{'}$ , where $a^{'}$ is the canonical image of $a \otimes_{A} A [T_{1}, \dots, T_{n}]$ , and is therefore an ideal of finite type of $A [S_{1}, \dots, S_{m}, T_{1}, \dots, T_{n}]$ ; the ideal $b$ of $B [T_{1}, \dots, T_{n}]$ is of the form $b^{'} / a^{'}$ , where $b^{'}$ is an ideal of $A [S_{1}, \dots, S_{m}, T_{1}, \dots, T_{n}]$ ; since $a^{'}$ and $b^{'} / a^{'}$ are modules of finite type over $A [S_{1}, \dots, S_{m}, T_{1}, \dots, T_{n}]$ , so is $b^{'}$ , and $C$ , isomorphic to $A [S_{1}, \dots, S_{m}, T_{1}, \dots, T_{n}] / b^{'}$ , is therefore an $A$ -algebra of finite presentation. Note finally that if $A$ is Noetherian, then so is $A [T_{1}, \dots, T_{n}]$ , so every $A$ -algebra of finite type is then of finite presentation.

Definition (1.4.2).

Let $f : X \to Y$ be a morphism of preschemes, $x$ a point of $X$ , $y = f (x)$ . We say that $f$ is of finite presentation at $x$ if there exist an affine open neighbourhood $V$ of $y$ and an affine open neighbourhood $U$ of $x$ such that $f (U) \subset V$ and the ring $A (U)$ is an $A (V)$ -algebra of finite presentation. We say that $f$ is locally of finite presentation if it is of finite presentation at every point of $X$ .

If $Y$ is locally Noetherian, to say that $f$ is locally of finite type and to say that $f$ is locally of finite presentation are equivalent.

Proposition (1.4.3).

(i) Every local isomorphism is locally of finite presentation.

(ii) If two morphisms $f : X \to Y$ , $g : Y \to Z$ are locally of finite presentation, then so is $g \circ f$ .

(iii) If $f : X \to Y$ is an $S$ -morphism locally of finite presentation, then $f_{(S^{'})} : X_{(S^{'})} \to Y_{(S^{'})}$ is locally of finite presentation for every extension $S^{'} \to S$ of the base prescheme.

(iv) If $f : X \to X^{'}$ and $g : Y \to Y^{'}$ are two $S$ -morphisms locally of finite presentation, then $f \times_{S} g$ is locally of finite presentation.

(v) If the composite $g \circ f$ of two morphisms is locally of finite presentation and if $g$ is locally of finite type, then $f$ is locally of finite presentation.

Assertion (i) is trivial. To prove (iii) one may restrict to the case where $S = Y$ (I, 3.3.11); let $x^{'}$ be a point of $X_{(S^{'})}$ , and $s^{'}$ , $x$ its projections onto $S^{'}$ and $X$ respectively, $s$ the common projection of $s^{'}$ and $x$ onto $S$ . By hypothesis there is an affine open neighbourhood $V$ (resp. $U$ ) of $s$ (resp. $x$ ) such that the image of $U$ is contained in $V$ and $A (U)$ is an $A (V)$ -algebra of finite presentation; let $V^{'}$ be an affine open neighbourhood of $s^{'}$ whose image in $S$ is contained in $V$ ; then $U \times_{V} V^{'}$ is an affine open neighbourhood of $x^{'}$ whose image in $S^{'}$ is contained in $V^{'}$ , and whose ring $A (U) \otimes_{A (V)} A (V^{'})$ is an $A (V^{'})$ -algebra of finite presentation (1.4.1).

To prove (ii), consider a point $x \in X$ , and let $y = f (x)$ , $z = g (f (x))$ . By hypothesis there is an affine open neighbourhood $W = Spec (C)$ (resp. $V = Spec (B)$ ) of $z$ (resp. $y$ ) such that $g (V) \subset W$ and $B$ is a $C$ -algebra of finite presentation. On the other hand, there is an affine open neighbourhood $V^{'} = Spec (B^{'})$ (resp. $U = Spec (A)$ ) of $y$ (resp. $x$ ) such that $f (U) \subset V^{'}$ and $A$ is a $B^{'}$ -algebra of finite presentation. Let $s \in B$ be such that the open set $D (s) = Spec (B_{s})$ in $V$ is a neighbourhood of $y$ contained in $V^{'}$ , and let $U^{'} = f^{- 1} (D (s)) \subset U$ ; one has $U^{'} = Spec (A \otimes_{B^{'}} B_{s})$ , and $A \otimes_{B^{'}} B_{s}$ is a $B_{s}$ -algebra of finite presentation (1.4.1); on the other hand, $B_{s} = B [T] / (1 - s T)$ is a $B$ -algebra of finite presentation by definition; consequently (1.4.1), $A \otimes_{B^{'}} B_{s}$ is a $C$ -algebra of finite presentation, which proves (ii).

One knows that (iv) results from (i), (ii), and (iii) (I, 3.5.1). To prove (v), consider the commutative diagram

  X        ──Γ_f──→  X ×_Z Y
  │                    │
  f│                  │f × 1
  ↓                    ↓
  Y        ──Δ_g──→  Y ×_Z Y

which identifies $X$ with the product of the $(Y \times_{Z} Y)$ -preschemes $Y$ and $X \times_{Z} Y$ (I, 5.3.7), $Δ_{g}$ being the diagonal morphism. If we know that $Δ_{g}$ is locally of finite presentation, it will follow from (iii) that the same is true of $Γ_{f}$ . On the other hand, one has the factorization of $f$ as $X Γ_{f} X \times_{Z} Y p_{2} Y$ (I, 5.3.13), where the projection $p_{2}$ is equal to $(g \circ f) \times_{Z} 1_{Y}$ ; since $g \circ f$ is supposed locally of finite presentation, so is $p_{2}$ by (iv), and it will finally result from (ii) that $f$ too is locally of finite presentation. One sees therefore that one is reduced to proving:

Corollary (1.4.3.1).

Let $g : Y \to Z$ be a morphism locally of finite type; then the diagonal morphism $Δ_{g} : Y \to Y \times_{Z} Y$ is locally of finite presentation.

One may restrict to the case where $Z = Spec (A)$ , $Y = Spec (B)$ , $B$ being an $A$ -algebra of finite type; the morphism $Δ_{g}$ then corresponds to the augmentation homomorphism $δ : B \otimes_{A} B \to B$ such that $δ (x \otimes y) = x y$ . In view of definition (1.4.2), it suffices to show that the kernel $J$ of $δ$ is an ideal of finite type, which results from $(0_{I V}, 20.4.4)$ .

Proposition (1.4.4).

Let $A$ be a ring, $B$ an $A$ -algebra, $B^{'}$ a polynomial algebra $A [T_{1}, \dots, T_{n}]$ , $u : B^{'} \to B$ a surjective homomorphism of $A$ -algebras. For $B$ to be an $A$ -algebra of finite presentation it is necessary and sufficient that the kernel $J$ of $u$ be an ideal of finite type of $B^{'}$ .

The condition is sufficient by definition (1.4.1). To see that it is necessary, remark that the morphism $g : Spec (B^{'}) \to Spec (A)$ is locally of finite type; if $B$ is an $A$ -algebra of finite presentation, the morphism $f : Spec (B) \to Spec (B^{'})$ corresponding to $u$ is such that $g \circ f : Spec (B) \to Spec (A)$ is locally of finite presentation, so it results from (1.4.3, (v)) that $f$ is also locally of finite presentation. Now, to show that the ideal $J$ is of finite type, it suffices to prove that $J$ is a $B^{'}$ -Module of finite type (II, 6.1.4.1). Returning to definition (1.4.2), one is finally reduced to proving:

Lemma (1.4.4.1).

Let $a$ be an ideal of a ring $C$ , $s$ an element of $C$ , such that $C_{s} / a_{s}$ is a $C$ -algebra of finite presentation; then $a_{s}$ is an ideal of finite type in the ring $C_{s}$ .

By hypothesis, there exist a polynomial $C$ -algebra $C^{'} = C [T_{1}, \dots, T_{n}]$ and a surjective $C$ -homomorphism $v : C^{'} \to C_{s} / a_{s}$ whose kernel $b$ is of finite type. Let $p : C_{s} \to C_{s} / a_{s}$ be the canonical homomorphism; for each $i$ there is a $t_{i} \in C$ such that $p (t_{i} / s^{k}) = v (T_{i})$ (one may suppose the exponent $k$ is the same for all indices $i$ ). Consider then the $C_{s}$ -algebra $D = C_{s} [T_{1}, \dots, T_{n}]$ , and the homomorphism of $C_{s}$ -algebras $w : D \to C_{s}$ such that $w (T_{i}) = t_{i} / s^{k}$ ; $w$ is manifestly surjective, and consequently so is $v^{'} = p \circ w : D \to C_{s} / a_{s}$ . On the other hand, every polynomial in $D$ may be written $P / s^{m}$ , where $P \in C^{'} = C [T_{1}, \dots, T_{n}]$ , and one has $v^{'} (P / s^{m}) = (1/ s^{m}) v (P)$ ; since $1/ s$ is invertible in $C_{s}$ , the relation $v^{'} (P / s^{m}) = 0$ in $C_{s} / a_{s}$ is equivalent to $v (P) = 0$ , and consequently the kernel $b^{'}$ of $v^{'}$ is generated by the canonical image of $b$ in $D$ , and a fortiori is of finite type in the ring $D$ . But $b^{'} = v^{' - 1} (0) = w^{- 1} (p^{- 1} (0)) = w^{- 1} (a_{s})$ , and since $w$ is surjective, $a_{s} = w (b^{'})$ , which proves that $a_{s}$ is an ideal of finite type of $C_{s}$ .

Corollary (1.4.5).

Let $X$ , $Y$ be two preschemes, $j : X \to Y$ an immersion, $U$ an open set of $Y$ such that $j (X)$ is closed in $U$ , $J$ the quasi-coherent Ideal of $O_{U}$ defining the closed subprescheme of $U$ associated with $j$ . For $j$ to be locally of finite presentation it is necessary and sufficient that $J$ be an $O_{U}$ -Module of finite type.

Proposition (1.4.6).

Let $ϕ : A \to B$ be a ring homomorphism. For the corresponding morphism $f : Spec (B) \to Spec (A)$ to be locally of finite presentation, it is necessary and sufficient that $B$ be an $A$ -algebra of finite presentation.

The condition being trivially sufficient, it remains to prove that it is necessary. If $f$ is locally of finite presentation, it already follows from (1.3.6) that $B$ is an $A$ -algebra of finite type, so there exists a surjective homomorphism $u : B^{'} = A [T_{1}, \dots, T_{n}] \to B$ of $A$ -algebras; it results from (1.4.3, (v)) that the immersion $j : Spec (B) \to Spec (B^{'})$ is locally

of finite presentation; therefore (1.4.5), if $J$ is the kernel of $u$ , the $B^{'}$ -Module $J$ is of finite type, and consequently (II, 6.1.4.1) $J$ is an ideal of finite type.

Proposition (1.4.7).

Let $ρ : A \to B$ be a ring homomorphism making $B$ into a finite $A$ -algebra. For $B$ to be an $A$ -algebra of finite presentation it is necessary and sufficient that $B$ be an $A$ -module of finite presentation.

We will use:

Lemma (1.4.7.1).

If $B$ is a finite $A$ -algebra, there exists a surjective $A$ -homomorphism of algebras $u : B^{'} \to B$ , where $B^{'}$ is a finite $A$ -algebra and of finite presentation which is a free $A$ -module.

Indeed, there is a finite system $(a_{i})_{1 \leq i \leq m}$ of generators of the $A$ -algebra $B$ such that each $a_{i}$ satisfies a relation $F_{i} (a_{i}) = 0$ , where $F_{i} \in A [X]$ is a unitary polynomial of degree > 0; the quotient $A$ -algebra $B_{i}^{'} = A [X] / (F_{i})$ is therefore free of finite rank over $A$ ; let $c_{i}$ be the class of $X$ in $B_{i}^{'}$ . There exists an $A$ -homomorphism of algebras $u_{i} : B_{i}^{'} \to B$ such that $u_{i} (c_{i}) = a_{i}$ ; it then suffices to take for $B^{'}$ the tensor product $B_{1}^{'} \otimes_{A} B_{2}^{'} \otimes \dots \otimes_{A} B_{m}^{'}$ , and to consider the homomorphism $u_{1} \otimes u_{2} \otimes \dots \otimes u_{m} : B^{'} \to B$ , which is surjective by construction. Moreover, if $B^{''} = A [T_{1}, \dots, T_{m}]$ and if $b_{i}^{'}$ denotes the canonical image of $c_{i}$ in $B^{'}$ , the $A$ -homomorphism of algebras $v : B^{''} \to B^{'}$ such that $v (T_{i}) = b_{i}^{'}$ ( $1 \leq i \leq m$ ) is surjective, and its kernel $b^{'}$ is the ideal of B'' generated by the $F_{i} (T_{i})$ , hence of finite type; this proves that $B^{'}$ is an $A$ -algebra of finite presentation.

This lemma being proved, keep the same notations and let $w = v \circ u : B^{''} \to B$ . If $b$ (resp. $a$ ) is the kernel of $w$ (resp. $u$ ), one has $a = v (b)$ since $v$ is surjective. Now (1.4.4), if $B$ is an $A$ -algebra of finite presentation, $b$ is an ideal of finite type of B'', so $a$ is an ideal of finite type of $B^{'}$ , hence an $A$ -module of finite type since $B^{'}$ is a finite $A$ -algebra; since $B^{'}$ is a free $A$ -module, $B$ is an $A$ -module of finite presentation. Conversely, if $B$ is an $A$ -module of finite presentation, $a$ is an $A$ -module of finite type (Bourbaki, Alg. comm., chap. I, §2, n° 8, lemme 9), and a fortiori an ideal of finite type of $B^{'}$ ; consequently, $B$ is by definition a $B^{'}$ -algebra of finite presentation, and since $B^{'}$ is an $A$ -algebra of finite presentation, $B$ is an $A$ -algebra of finite presentation.

1.5. Morphisms of finite type

Proposition (1.5.1).

Let $f : X \to Y$ be a morphism, $(U_{α})$ a cover of $Y$ formed of affine open sets. The following conditions are equivalent:

a) $f$ is locally of finite type and quasi-compact.

b) For every $α$ , $f^{- 1} (U_{α})$ is a finite union of affine open sets $V_{α i}$ such that each ring $A (V_{α i})$ is an $A (U_{α})$ -algebra of finite type.

c) For every affine open set $U$ of $Y$ , $f^{- 1} (U)$ is a finite union of affine open sets $V_{i}$ such that the rings $A (V_{i})$ are $A (U)$ -algebras of finite type.

For the proof, see (I, 6.3.2 and 6.6.3).

Definition (1.5.2).

We say that a morphism $f$ is of finite type if it satisfies the equivalent conditions of proposition (1.5.1).

Proposition (1.5.3).

Let $f : X \to Y$ be a morphism of finite type; if $Y$ is Noetherian, then so is $X$ .

For the proof, see (I, 6.3.7).

Proposition (1.5.4).

(i) Let $j : X \to Y$ be a morphism of immersion. If $j$ is quasi-compact (which is the case if $j$ is a closed immersion, or if the space underlying $X$ is Noetherian, or if the space underlying $Y$ is locally Noetherian), then $j$ is of finite type.

(ii) The composite of two morphisms of finite type is of finite type.

(iii) If $f : X \to Y$ is an $S$ -morphism of finite type, then $f_{(S^{'})} : X_{(S^{'})} \to Y_{(S^{'})}$ is of finite type for every extension $S^{'} \to S$ of the base prescheme.

(iv) If $f : X \to X^{'}$ and $g : Y \to Y^{'}$ are two $S$ -morphisms of finite type, then $f \times_{S} g$ is of finite type.

(v) If the composite $g \circ f$ of two morphisms is of finite type, and if $g$ is quasi-separated or $X$ is Noetherian, then $f$ is of finite type.

(vi) If a morphism $f$ is of finite type, then so is $f_{re d}$ .

This results at once (except case (v) where $X$ is Noetherian, proved in (I, 6.3.6)) from the definitions and from (1.1.2), (1.2.4), and (1.3.4).

Corollary (1.5.5).

Let $f : X \to Y$ be a morphism of finite type. For every morphism $Y^{'} \to Y$ such that $Y^{'}$ is Noetherian, $X \times_{Y} Y^{'}$ is Noetherian.

This results from (1.5.4, (iii)) and from (1.5.3).

Corollary (1.5.6).

Let $X$ be a prescheme of finite type over a locally Noetherian prescheme $S$ . Then every $S$ -morphism $f : X \to Y$ is of finite type.

For the proof, see (I, 6.3.9).

Proposition (1.5.7).

Let $ϕ : A \to B$ be a ring homomorphism. For the corresponding morphism $f : Spec (B) \to Spec (A)$ to be of finite type, it is necessary and sufficient that $ϕ$ make $B$ into an $A$ -algebra of finite type.

For the proof, see (I, 6.3.3).

1.6. Morphisms of finite presentation

Definition (1.6.1).

Let $X$ , $Y$ be two preschemes. We say that a morphism $f : X \to Y$ is of finite presentation if it satisfies the three following conditions:

(i) $f$ is locally of finite presentation;

(ii) $f$ is quasi-compact (which, when (i) is satisfied, is equivalent to saying that $f$ is of finite type (1.5.2));

(iii) $f$ is quasi-separated.

We say in this case that $X$ is of finite presentation over $Y$ , or a $Y$ -prescheme of finite presentation.

Condition (iii) is automatically satisfied if $f$ is separated, or if $X$ is locally Noetherian (1.2.8); when $Y$ is locally Noetherian, to say that $f$ is of finite presentation and to say that $f$ is of finite type are equivalent (the latter condition implying that $X$ is locally Noetherian (1.3.3)).

Proposition (1.6.2).

(i) Every quasi-compact immersion locally of finite presentation (in particular, every quasi-compact open immersion) is of finite presentation.

(ii) The composite of two morphisms of finite presentation is of finite presentation.

(iii) Let $X$ , $Y$ be two $S$ -preschemes, $f : X \to Y$ an $S$ -morphism of finite presentation. Then, for every morphism $S^{'} \to S$ , the morphism $f_{(S^{'})} : X_{(S^{'})} \to Y_{(S^{'})}$ is of finite presentation.

(iv) Let $f : X \to Y$ , $f^{'} : X^{'} \to Y^{'}$ be two $S$ -morphisms of finite presentation; then $f \times_{S} f^{'} : X \times_{S} X^{'} \to Y \times_{S} Y^{'}$ is of finite presentation.

(v) If the composite $g \circ f$ of two morphisms is of finite presentation, and if $g$ is quasi-separated and locally of finite presentation, then $f$ is of finite presentation.

This results immediately from (1.1.2), (1.2.2), (1.2.4), and (1.4.3).

It follows in particular from (1.6.2, (iii)) that if $f$ is a morphism of finite presentation and $U$ an open set of $Y$ , the restriction $f^{- 1} (U) \to U$ of $f$ is a morphism of finite presentation. Conversely, let $(U_{α})$ be a cover of $Y$ by affine open sets, and suppose that the restrictions $f^{- 1} (U_{α}) \to U_{α}$ of $f$ are morphisms of finite presentation; it then follows that $f$ is of finite presentation, since $f$ is manifestly locally of finite presentation, quasi-compact by virtue of (1.1.1), and quasi-separated by virtue of (1.2.6). One may therefore say that the property of a morphism $f : X \to Y$ of being of finite presentation is local on $Y$ .

If $X$ is a quasi-separated prescheme, every morphism $f : X \to Y$ is quasi-separated (1.2.3). Therefore, if $f$ is quasi-compact and locally of finite presentation, $f$ is then of finite presentation.

In particular:

Corollary (1.6.3).

Let $ϕ : A \to B$ be a ring homomorphism. For the corresponding morphism $f : Spec (B) \to Spec (A)$ to be of finite presentation, it is necessary and sufficient that $ϕ$ make $B$ into an $A$ -algebra of finite presentation.

The necessity of the condition results from (1.4.6).

Remark (1.6.4).

In definition (1.6.1), condition (iii) is not a consequence of the two others. Let, for example, $Y$ be an affine scheme whose underlying space is not Noetherian, and let $U$ be a non-quasi-compact open set in $Y$ . Let $X$ be the prescheme obtained by gluing two preschemes Y_1, Y_2 isomorphic to $Y$ along the open sets U_1, U_2 corresponding to $U$ $(0_{I}, 4.1.7)$ , so that $X$ is a union of two open sets isomorphic respectively to Y_1 and Y_2 (and which we identify with these), with $Y_{1} \cap Y_{2} = U$ . In addition, there is a morphism $f : X \to Y$ that coincides on $Y_{i}$ with the given isomorphism $Y_{i} \to Y$ ( $i = 1, 2$ ). It is clear that $f$ satisfies condition (i) of (1.6.1), being a local isomorphism; it also satisfies (ii), the inverse image of a quasi-compact open set of $Y$ being the union of its images in Y_1 and Y_2; but as $Y_{1} \cap Y_{2}$ is not quasi-compact, $f$ is not quasi-separated (1.2.7).

Proposition (1.6.5).

Let $X^{'}$ , X'' be two preschemes, $X = X^{'} ⨿ X^{''}$ their sum, $f^{'} : X^{'} \to Y$ , $f^{''} : X^{''} \to Y$ two morphisms, $f : X \to Y$ the morphism that coincides with $f^{'}$ on $X^{'}$ and with f'' on X''. For $f$ to be of finite presentation it is necessary and sufficient that $f^{'}$ and f'' be so.

It suffices to show that for $f$ to possess one of the three properties of definition

(1.6.1), it is necessary and sufficient that $f^{'}$ and f'' possess it; this is clear for the property of being locally of finite presentation, which is local on $X$ ; for the property of being quasi-compact, this was seen in (1.1.6), and for the property of being quasi-separated, in (1.2.9).

1.7. Improvements of earlier results

We give in this number a list of propositions proved in the preceding chapters whose statement may be improved by means of the new finiteness conditions introduced above.

(1.7.1) In the statements of (I, 6.4.2, 6.4.3, and 6.4.9), one may, instead of supposing that $X$ and $Y$ are of finite type over the field $K$ , suppose only that they are locally of finite type over $K$ ; for (I, 6.4.2), it suffices to observe that every point of a prescheme $X$ which is closed in every affine open set of $X$ is closed in $X$ , and an affine scheme locally of finite type over $K$ is ipso facto of finite type (1.5.1). For (I, 6.4.9), one uses (1.3.7) and (1.3.4, (v)).

(1.7.2) In (I, 6.5.1, (ii)), one may, instead of supposing that $S$ is locally Noetherian and $Y$ of finite type over $S$ , suppose only that $Y$ is locally of finite presentation over $S$ , as the proof immediately shows (applying definition (1.4.2)). Similarly, in (I, 6.5.4, (ii)) and (I, 6.5.5, (ii)) it suffices to suppose that $f$ is an $S$ -morphism, $Y$ being locally of finite presentation over $S$ .

(1.7.3) In (I, 7.1.11), for the second assertion, instead of supposing $S$ locally Noetherian and $Y$ of finite type over $S$ , one may suppose only $Y$ locally of finite presentation over $S$ (the proof depending on (I, 6.5.1, (ii))); similarly in (I, 7.1.12 to 7.1.15).

(1.7.4) In (I, 9.2.2), one may replace the three hypotheses by the single hypothesis (entailed by each of the others) that $f$ is quasi-compact and quasi-separated, by virtue of (1.2.6) and (1.2.7). Note that in (I, 9.2.1), the hypothesis means exactly that $f$ is quasi-compact and quasi-separated.

(1.7.5) In (I, 9.3.1, 9.3.2, and 9.3.3), one may replace the two hypotheses on $X$ by the hypothesis (less restrictive) that $X$ is a quasi-compact and quasi-separated prescheme, the proof using exactly these two properties (by virtue of (1.2.7)).

(1.7.6) In (I, 9.3.4 and 9.3.5), it suffices to suppose $X$ quasi-compact, and $F$ and $J$ quasi-coherent and of finite type.

(1.7.7) In (I, 9.4.7), one may weaken hypothesis b) by supposing only $X$ quasi-separated, since it suffices only that the canonical immersion $U \to X$ be quasi-compact (1.2.7). One may make the same modification in (I, 9.4.8, 9.4.9, and 9.4.10).

(1.7.8) In (I, 9.5.1), the conditions indicated in parentheses in the statement may be replaced by the condition that $f$ be quasi-compact and quasi-separated.

(1.7.9) In (I, 9.6.5), one may replace the hypotheses on $X$ by the less restrictive hypothesis that $X$ is quasi-compact and quasi-separated, as the proof shows, since it

uses only (I, 9.4.7). In (I, 9.6.6), the hypothesis " $X$ is a quasi-compact scheme" may be replaced by " $X$ is a quasi-compact and quasi-separated prescheme".

(1.7.10) In (II, 1.7.15), one may suppose only that $Y$ is quasi-compact and quasi-separated, the proof using only (I, 9.6.5).

(1.7.11) In the second assertion of (II, 3.4.5), one may substitute for the two hypotheses on $Y$ the weaker hypothesis that $Y$ is quasi-compact and quasi-separated. Similarly in (II, 3.4.8).

(1.7.12) In (II, 3.8.5), one may likewise suppose only that $Y$ is quasi-compact and quasi-separated, this hypothesis intervening through (I, 9.4.7).

(1.7.13) In (II, 4.4.1, 4.4.6, and 4.4.7), it suffices to suppose that $Y$ is quasi-compact and quasi-separated, taking account of (1.2.3) in the proof of (II, 4.4.7). Similarly, in (II, 4.4.10.1), it suffices to suppose $Z$ quasi-compact and quasi-separated.

(1.7.14) In (II, 4.5.2) and (II, 4.5.5), it suffices to suppose $X$ quasi-compact and quasi-separated, this hypothesis intervening through (I, 9.3.1 and 9.3.2).

(1.7.15) In (II, 4.6.8), it again suffices to suppose $X$ quasi-compact and quasi-separated, this hypothesis intervening through (I, 9.4.7).

(1.7.16) In (II, 5.1.2), it suffices to suppose that $X$ is quasi-compact and quasi-separated (the hypothesis intervening through (II, 4.5.2 and 4.5.5)). Similarly in (II, 5.1.9), where one uses (I, 9.6.5).

(1.7.17) In (II, 5.2.1), it again suffices to suppose $X$ quasi-compact and quasi-separated (the hypothesis intervening through (II, 4.5.2)).

(1.7.18) In (II, 5.2.2), one must suppose $f$ quasi-compact and $X$ quasi-separated; the present proof is in fact insufficient (with the hypotheses made), for from the fact that for every quasi-coherent $O_{X}$ -Module $F$ one has $R^{1} f_{*} (F) = 0$ , it does not necessarily follow, for an affine open set $U$ of $Y$ , that the same property holds for the quasi-coherent $(O_{X} ∣ f^{- 1} (U))$ -Modules, if one does not know that such a Module is the restriction of a quasi-coherent $O_{X}$ -Module (the same remark applying to conditions b) and c')); when $f$ is quasi-compact and $X$ quasi-separated, this extension is possible by virtue of (I, 9.4.7), modified above (1.7.7).

(1.7.19) In (II, 5.3.2), it suffices to suppose $Y$ quasi-compact and quasi-separated (which intervenes through (II, 4.4.6 and 4.4.7)). Similarly in (II, 5.5.3, (ii)).

(1.7.20) In (III, 1.2.2, 1.2.3, and 1.2.4), one may replace the two hypotheses on $X$ by the less restrictive hypothesis that $X$ is a quasi-compact and quasi-separated prescheme, the proof using only (I, 9.3.3).

(1.7.21) In (III, 1.4.10 to 1.4.14), one may replace "separated" by "quasi-separated", this hypothesis serving only to allow application of (I, 9.2.2) (see above (1.7.4)). Similarly, in (III, 1.4.15), it suffices to suppose that the morphism $u$ is quasi-separated and quasi-compact.

(1.7.22) In (III, 2.1.3), one may again replace the hypotheses by the less restrictive hypothesis that $X$ is a quasi-compact and quasi-separated prescheme, the reasoning being the same as in (III, 1.2.2).

1.8. Morphisms of finite presentation and constructible sets

(1.8.1)

Let $X$ be a prescheme; one knows that every quasi-compact open set in $X$ is a finite union of affine open sets, and conversely. For an open set $U$ of $X$ to be retrocompact in $X$ $(0_{III}, 9.1.1)$ , it is therefore necessary and sufficient that for every affine open set $V$ of $X$ , $U \cap V$ be quasi-compact. When $X$ is quasi-separated (1.2.1), every quasi-compact open set in $X$ is retrocompact in $X$ (1.2.7), so every locally constructible part of $X$ is retrocompact in $X$ $(0_{III}, 9.1.13)$ ; if moreover $X$ is quasi-compact, there is identity between constructible parts and locally constructible parts in $X$ $(0_{III}, 9.1.12)$ , and between open constructible parts and open quasi-compact parts $(0_{III}, 9.1.5)$ .

Proposition (1.8.2).

Let $X$ , $Y$ be two preschemes, $f : X \to Y$ a morphism. For every constructible (resp. locally constructible) part $Z$ of $Y$ , $f^{- 1} (Z)$ is a constructible (resp. locally constructible) part of $X$ .

Suppose $Z$ locally constructible; for every $x \in X$ there will be an affine open neighbourhood $V$ of $f (x)$ such that $Z \cap V$ is constructible in $V$ ; if the proposition is proved for constructible parts, $f^{- 1} (Z) \cap f^{- 1} (V)$ will be constructible in $f^{- 1} (V)$ , so $f^{- 1} (Z)$ will be locally constructible. It therefore suffices to consider the case where $Z$ is constructible, and, taking account of $(0_{III}, 9.1.3)$ , one is reduced to the case where $Z$ is open and retrocompact in $Y$ , that is, such that the canonical injection $Z \to Y$ is a quasi-compact morphism; it then suffices to see that $f^{- 1} (Z)$ is retrocompact in $X$ , that is, such that the canonical injection $f^{- 1} (Z) \to X$ is a quasi-compact morphism; but this results from (1.1.2, (iii)), since $f^{- 1} (Z) = Z \times_{Y} X$ (I, 4.4.1).

Lemma (1.8.3).

Let $X$ be a quasi-compact and quasi-separated prescheme, $Z$ a constructible part of $X$ . There then exist an affine scheme $X^{'}$ and a morphism of finite presentation $f : X^{'} \to X$ such that $f (X^{'}) = Z$ .

As $X$ is quasi-compact, it is a finite union of affine open sets $X_{i}$ ( $i \in I$ ), and as $X$ is quasi-separated, it follows from (1.2.7) that each of the canonical immersions $g_{i} : X_{i} \to X$ is of finite presentation; one concludes that if $f_{i} : X_{i}^{'} \to X_{i}$ is a morphism of finite presentation, so is $g_{i} \circ f_{i} : X_{i}^{'} \to X$ (1.6.2), and if $X^{'}$ is the sum prescheme of the $X_{i}^{'}$ , the morphism $h : X^{'} \to X$ coinciding with $g_{i} \circ f_{i}$ on each $X_{i}^{'}$ is also of finite presentation (1.6.5). As $Z \cap X_{i}$ is constructible in $X_{i}$ $(0_{III}, 9.1.8)$ , one sees that one is reduced to proving the lemma when $X_{i}$ is affine of ring $A$ . Since $Z$ is then a finite union of sets of the form $U \cap ∁ V$ , where $U$ and $V$ are quasi-compact open sets $(0_{III}, 9.1.3)$ , one sees, by considering again a suitable sum of preschemes, that one may reduce to the case where $Z = U \cap ∁ V$ ; moreover, since $U$ is a finite union of affine open sets, one may even restrict to the case where $U$ is affine. Since $V$ is quasi-compact, it is a finite union of open sets of the form $X_{f_{i}}$ , where $f_{i} \in A$ ; let $J$ be the ideal of $A$ generated by the $f_{i}$ , and let $Z^{'}$ be the closed affine subscheme of $X$ that it defines (and which is by construction of finite presentation over $X$ ); by definition $V = ∁ Z^{'}$ , so $U \cap ∁ V = U \cap Z^{'}$ . Consider the affine scheme $X^{'} = Z^{'} \times_{X} U$ ,

and let $f : X^{'} \to X$ be the structure morphism; one has just seen that the canonical immersion $Z^{'} \to X$ is of finite presentation, and the same is true of the open immersion $U \to X$ , which is quasi-compact since $U$ and $X$ are affine; one concludes therefore from (1.6.2, (iv)) that $f$ is of finite presentation, and one has $f (X^{'}) = Z^{'} \cap U$ (I, 3.4.8), which completes the proof.

Theorem (1.8.4) (Chevalley).

Let $f : X \to Y$ be a morphism of finite presentation. For every locally constructible part $Z$ of $X$ , $f (Z)$ is locally constructible in $Y$ .

Let $y \in Y$ and $V$ an affine open neighbourhood of $y$ ; since $f$ is quasi-compact and quasi-separated, so is its restriction $f^{- 1} (V) \to V$ , hence $f^{- 1} (V)$ is a quasi-compact and quasi-separated prescheme (1.2.3); as $Z \cap f^{- 1} (V)$ is constructible in $f^{- 1} (V)$ (1.8.1), one sees that one may restrict to the case where $Y$ is affine and $Z$ constructible; $X$ is then quasi-compact and quasi-separated, so (1.8.3) there exists a morphism of finite presentation $g : X^{'} \to X$ such that $Z = g (X^{'})$ ; since $f \circ g$ is of finite presentation, one sees that one is reduced to the case where $Z = X$ ; in other words, it will suffice to prove:

Lemma (1.8.4.1).

Let $Y$ be an affine scheme, $f : X \to Y$ a quasi-compact morphism locally of finite presentation; then $f (X)$ is a constructible part of $Y$ .

As $X$ is quasi-compact, it is a finite union of affine open sets; one may therefore restrict to the case where $Y = Spec (A)$ , $X = Spec (B)$ , $B$ being an $A$ -algebra of finite presentation (1.4.6). Now, one has:

Lemma (1.8.4.2).

Let A_0 be a ring, $(A_{λ})_{λ \in L}$ an inductive system of A_0-algebras, $A = lim A_{λ}$ ; if $B$ is an $A$ -algebra of finite presentation, there exist a $λ$ and an $A_{λ}$ -algebra of finite presentation $B_{λ}$ such that $B$ is isomorphic to $B_{λ} \otimes_{A_{λ}} A$ .

By hypothesis, $B$ is isomorphic to an algebra of the form $A [T_{1}, \dots, T_{n}] / J$ , where the $T_{i}$ are indeterminates and $J$ an ideal of finite type; let $(F_{j})$ ( $1 \leq j \leq m$ ) be a system of generators of $J$ . Let $ϕ_{λ} : A_{λ} \to A$ be the canonical homomorphism. As $L$ is filtered, there exists $λ$ such that each of the coefficients of each of the polynomials $F_{j}$ is the image under $ϕ_{λ}$ of an element of $A_{λ}$ . In other words, there exists a system of polynomials $(F_{jλ})_{1 \leq j \leq m}$ of $A_{λ} [T_{1}, \dots, T_{n}]$ such that $F_{j} = ϕ_{λ} (F_{jλ})$ for $1 \leq j \leq m$ . If $J_{λ}$ is the ideal of $A_{λ} [T_{1}, \dots, T_{n}]$ generated by the $F_{jλ}$ , $J$ is the image of $J_{λ} \otimes_{A_{λ}} A$ in $A [T_{1}, \dots, T_{n}] = A_{λ} [T_{1}, \dots, T_{n}] \otimes_{A_{λ}} A$ ; the ring $B_{λ} = A_{λ} [T_{1}, \dots, T_{n}] / J_{λ}$ answers the question.

One will apply this lemma here by considering $A$ as the inductive limit of its sub- $Z$ -algebras of finite type. One sees therefore that there exists such a sub- $Z$ -algebra A_0 of $A$ , and an A_0-algebra of finite presentation B_0 such that $B$ is isomorphic to $B_{0} \otimes_{A_{0}} A$ ; set $X_{0} = Spec (B_{0})$ , $Y_{0} = Spec (A_{0})$ , so that $X = X_{0} \times_{Y_{0}} Y$ , $f$ being the projection $X \to Y$ ; let $f_{0} : X_{0} \to Y_{0}$ , $g_{0} : Y \to Y_{0}$ be the structure morphisms; it follows from (I, 3.4.8) that one has $f (X) = g_{0}^{- 1} (f_{0} (X_{0}))$ ; taking account of (1.8.2), it therefore suffices to show that $f_{0} (X_{0})$ is constructible. In other words, one is finally reduced to proving:

Corollary (1.8.5).

Let $Y$ be a Noetherian prescheme, $f : X \to Y$ a morphism of finite type. For every constructible part $Z$ of $X$ , $f (Z)$ is a constructible part of $Y$ .

One reduces as above to proving that $f (X)$ is constructible. Apply criterion $(0_{III}, 9.2.3)$ : it suffices to prove that for every irreducible closed part $T$ of $Y$ , $T \cap f (X) = f (f^{- 1} (T))$ is either rare in $T$ or contains a non-empty open of $T$ ; taking a subprescheme of $Y$ having $T$ as underlying space (I, 5.2.1) and considering its inverse image under $f$ , one sees finally (taking account of (1.5.4, (iii))) that one is reduced to proving that if one supposes $Y$ irreducible and $f$ dominant, then $f (X)$ contains a non-empty open of $Y$ . One may further suppose $Y$ affine; then $X$ is a finite union of affine open sets $V_{i}$ , and as $f (X)$ is dense in $Y$ , the same is true of at least one of the $f (V_{i})$ . One may therefore also suppose $X$ affine; if $(X_{j})$ is the (finite) family of irreducible components of $X$ , one sees as above that at least one of the $f (X_{j})$ is dense in $Y$ ; one may therefore suppose $X$ irreducible. Finally, replacing $f$ by $f_{re d}$ (1.5.4, (vi)), one may suppose $X$ and $Y$ integral. Then $X = Spec (B)$ , $Y = Spec (A)$ , where $A$ is a Noetherian integral ring, $B$ an integral $A$ -algebra of finite type containing $A$ (I, 1.2.7). It suffices to show that there exists $g \in A$ such that, for every $y \in D (g)$ (that is, such that $g \in / j_{y}$ ), the ideal $j_{y}$ is the intersection of $A$ and a prime ideal of $B$ , for this will show that $D (g) \subset f (X)$ . Finally, it suffices to prove:

Lemma (1.8.5.1).

Let $A$ be an integral ring, $B$ an integral $A$ -algebra of finite type. There exists $g \in A$ such that every homomorphism of $A$ into an algebraically closed field $Ω$ , non-zero on $g$ , extends to a homomorphism of $B$ into $Ω$ .

Now, this is a classical result of commutative algebra (Bourbaki, Alg. comm., chap. V, §3, n° 1, cor. 3 du th. 1).

Corollary (1.8.6).

Let $Y$ be an irreducible prescheme, $η$ its generic point, $f : X \to Y$ a morphism locally of finite type. If $f^{- 1} (η)$ is not empty, there exists an open neighbourhood $U$ of $η$ in $Y$ such that $f^{- 1} (y)$ is non-empty for every $y \in U$ . If $F$ is a quasi-coherent $O_{X}$ -Module of finite type, and if $F_{η} = F \otimes_{O_{Y}} k (η)$ is not zero, then there exists an open neighbourhood $U^{'}$ of $η$ in $Y$ such that $F_{y} = F \otimes_{O_{Y}} k (y)$ is not zero for every $y \in U^{'}$ .

If $p_{y}$ is the canonical projection of the fibre $f^{- 1} (y) = X \times_{Y} Spec (k (y))$ into $X$ , one has $F_{y} = p_{y}^{*} (F)$ , so Supp(ℱ_y) = p_y⁻¹(Supp(ℱ)) = Supp(ℱ) ∩ f⁻¹(y) by virtue of (I, 9.1.13.1) and (I, 3.6.1), since $F$ is of finite type; furthermore $S u pp (F)$ is closed in $X$ $(0_{I}, 5.2.2)$ , and if $Z$ is a closed subprescheme of $X$ having $S u pp (F)$ as underlying space (I, 5.2.1), and $j$ the canonical immersion $Z \to X$ , $f \circ j$ is locally of finite type (1.3.4); this shows that the first assertion entails the second. To prove the first assertion, remark first that one may suppose $X$ and $Y$ reduced (1.3.4, (vi)), that is, $Y$ integral. Let $ξ$ be a point of $f^{- 1} (η)$ ; one may replace $X$ and $Y$ by affine open neighbourhoods of $ξ$ and $η$ respectively, and hence suppose that $X = Spec (B)$ , $Y = Spec (A)$ , $B$ being an $A$ -algebra of finite type. In addition, if $Z$ is the reduced closed subprescheme of $X$ having $\overline{ξ}$ as underlying set (I, 5.2.1), one may, as above, replace $X$ by $Z$ , that is, suppose $B$ integral (I, 5.1.4). As the morphism $f$ is then dominant, the corresponding homomorphism $ϕ : A \to B$ is injective (I, 1.2.7); so the corollary results finally from lemma (1.8.5.1).

Proposition (1.8.7).

Let $S$ be a prescheme, $g : X \to S$ , $h : Y \to S$ two morphisms of finite presentation, $f : X \to Y$ an $S$ -morphism. For every $s \in S$ , set $X_{s} = g^{- 1} (s) = X \times_{S} Spec (k (s))$ , $Y_{s} = h^{- 1} (s) = Y \times_{S} Spec (k (s))$ , $f_{s} = f \times 1 : X_{s} \to Y_{s}$ . Then the set of $s \in S$ such that $f_{s}$ is surjective (resp. radicial) is locally constructible.

Let $E$ be the set of $s \in S$ such that $f_{s}$ is surjective; the set $S - E$ is equal to $h (Y - f (X))$ ; now $f$ is of finite presentation (1.6.2, (v)), so $f (X)$ is locally constructible in $Y$ (1.8.4); since $h$ is of finite presentation, $h (Y - f (X))$ is locally constructible in $S$ , hence so is $E$ .

To show that the set $E^{'}$ of $s \in S$ such that $f_{s}$ is radicial is locally constructible, we will use:

Lemma (1.8.7.1).

Let $U$ , $V$ be two preschemes; for a morphism $f : U \to V$ to be radicial, it is necessary and sufficient that the diagonal morphism $Δ_{f} : U \to U \times_{V} U$ be surjective. Consequently, every radicial morphism is separated.

Indeed, $Δ_{f}$ , being an immersion (I, 5.3.9), is a morphism locally of finite type (1.3.4); to say that $Δ_{f}$ is surjective therefore means that for every algebraically closed field $K$ , the corresponding map $U (K) \to (U \times_{V} U) (K) = U (K) \times_{V (K)} U (K)$ is surjective (1.3.7 and I, 3.4.2.1); but by definition of a fibre product of sets, this means that the map $U (K) \to V (K)$ corresponding to $f$ is injective, and this is equivalent to saying that $f$ is radicial (I, 3.5.5).

This being so, $Δ_{f_{s}}$ is deduced from $Δ_{f} : X \to X \times_{Y} X$ by the base change $Spec (k (s)) \to S$ (I, 5.3.4). Since $f$ is of finite presentation, so is the structure morphism $X \times_{Y} X \to Y$ (1.6.2, (iv)), hence also $Δ_{f}$ (1.6.2, (v)); it therefore suffices to apply the first part of the proposition to $Δ_{f}$ , using lemma (1.8.7.1).

1.9. Pro-constructible and ind-constructible sets

Lemma (1.9.1).

Let $(A_{α})_{α \in I}$ be an inductive system of rings whose index set $I$ is filtered to the right, and let $A = lim A_{α}$ . For $A = 0$ it is necessary and sufficient that there exist an index $γ$ such that $A_{γ} = 0$ (and one then has $A_{β} = 0$ for every $β \geq γ$ ).

Indeed, for every $α \in I$ , the canonical homomorphism $ϕ_{α} : A_{α} \to A$ transforms the unit element into the unit element; to say that $A = 0$ means that $ϕ_{α} (1) = 0$ , or again that $ϕ_{α} (1) = ϕ_{α} (0)$ ; one knows that this is equivalent to saying that there exists an index $γ \geq α$ such that the homomorphism $ϕ_{γ α} : A_{α} \to A_{γ}$ is such that $ϕ_{γ α} (1) = ϕ_{γ α} (0)$ , and this last relation is equivalent to $A_{γ} = 0$ , whence the lemma.

Proposition (1.9.2).

Let $B$ be a ring, $(A_{α})_{α \in I}$ an inductive system of $B$ -algebras whose index set $I$ is filtered to the right, and let $A = lim A_{α}$ ; set $Y = Spec (B)$ , $X_{α} = Spec (A_{α})$ , $X = Spec (A)$ , and let $u : X \to Y$ , $u_{α} : X_{α} \to Y$ be the morphisms corresponding to the structure homomorphisms $B \to A$ , $B \to A_{α}$ . Then:

(i) For $X = \emptyset$ it is necessary and sufficient that there exist a $γ \in I$ such that $X_{γ} = \emptyset$ , and one then has $X_{α} = \emptyset$ for $α \geq γ$ .

(ii) One has

  u(X) = ⋂_{α ∈ I} u_α(X_α).                                                    (1.9.2.1)

Assertion (i) is nothing but the translation of (1.9.1). To prove (ii), note that, since $u$ factors as $X \to X_{α} u_{α} Y$ , the first member of (1.9.2.1) is contained in the second. Conversely, let $y \in Y - u (X)$ ; one has $u^{- 1} (y) = \emptyset$ , and $u^{- 1} (y)$ is the space underlying the $k (y)$ -prescheme $Spec (A \otimes_{B} k (y))$ ; as in the category of $B$ -modules the functor $lim$ commutes with the tensor product, $A \otimes_{B} k (y)$ is the inductive limit of the inductive system of rings $A_{α} \otimes_{B} k (y)$ ; lemma (1.9.1) shows that there exists $α$ such that $A_{α} \otimes_{B} k (y) = 0$ , that is, $u_{α}^{- 1} (y) = \emptyset$ , hence $y \in / u_{α} (X_{α})$ . C.Q.F.D.

Proposition (1.9.3).

Let $X$ be a quasi-compact and quasi-separated prescheme, $E$ a part of $X$ , $(X_{α})_{α \in I}$ an affine open cover of $X$ , and for every $α \in I$ , set $E_{α} = E \cap X_{α}$ . The following conditions are equivalent:

a) There exists a quasi-compact morphism $f : X^{'} \to X$ such that $E = f (X^{'})$ .

a') For every $α$ , there exists a quasi-compact morphism $f_{α} : X_{α}^{'} \to X_{α}$ such that $E_{α} = f_{α} (X_{α}^{'})$ .

a'') For every $α$ , there exist an affine scheme $X_{α}^{'}$ and a morphism $f_{α} : X_{α}^{'} \to X_{α}$ such that $f_{α} (X_{α}^{'}) = E_{α}$ .

b) $E$ is an intersection of constructible parts of $X$ .

b') $F = X - E$ is a union of constructible parts of $X$ .

It is clear that b) and b') are equivalent. Condition a) entails a') by taking for $X_{α}^{'}$ the prescheme induced on the open set $f^{- 1} (X_{α})$ of $X^{'}$ and for $f_{α}$ the restriction of $f$ . To see that a') entails a''), it suffices to remark that $X_{α}^{'}$ is a union of affine open sets $X_{α λ}^{''}$ ( $λ \in L_{α}$ ); let $X_{α}^{''}$ be the sum prescheme of the $X_{α λ}^{''}$ ( $λ \in L_{α}$ ), which is an affine scheme; if $g_{α} : X_{α}^{''} \to X_{α}^{'}$ is the morphism coinciding with the identity on each of the $X_{α λ}^{''}$ , it is clear that if one sets $f_{α}^{'} = f_{α} \circ g_{α}$ , one has $E_{α} = f_{α}^{'} (X_{α}^{''})$ since $g_{α}$ is surjective. To show that a'') entails a), note that there is a finite subset $J$ of $I$ such that the $X_{α}$ of indices $α \in J$ cover $X$ ; let $X^{'}$ be the sum prescheme of the $X_{α}^{'}$ for $α \in J$ , and let $f : X^{'} \to X$ be the morphism coinciding with $j_{α} \circ f_{α}$ for every $α \in J$ , where $j_{α} : X_{α} \to X$ is the canonical injection. As $E$ is the union of the $E \cap X_{α}$ for $α \in J$ , one has indeed $E = f (X^{'})$ , and it remains to see that $f$ is a quasi-compact morphism; but the hypothesis that $X$ is quasi-separated entails that $j_{α}$ is quasi-compact (1.2.7), hence so is $j_{α} \circ f_{α}$ (1.1.1 and 1.1.2) and finally $f$ (1.1.6).

It remains then to prove the equivalence of a'') and b). Let us prove first that a'') entails b): consider a finite subset $J$ of $I$ such that the $X_{α}$ for $α \in J$ cover $X$ ; it will suffice to show that, for every $α \in J$ , $E_{α}$ is an intersection of constructible parts $F_{α λ}$ of $X_{α}$ ( $λ \in L_{α}$ ); indeed, every $F_{α λ}$ is also a constructible part of $X$ by virtue of $(0_{III}, 9.1.8, (ii))$ , because the hypothesis that $X$ is quasi-separated entails that every $X_{α}$ is retrocompact in $X$ (1.2.7); $E$ being the union of the $E_{α}$ for $α \in J$ , is intersection of the (finite) unions $⋃_{α \in J} F_{α, λ (α)}$ , for all choices of $λ (α) \in L_{α}$ ; these unions being constructible in $X$ , this establishes our assertion. One may therefore restrict to the case

where $X = Spec (A)$ and where $E = f (X^{'})$ , $X^{'} = Spec (A^{'})$ being affine, $f : X^{'} \to X$ a morphism corresponding to a ring homomorphism $A \to A^{'}$ . Note now that $A^{'}$ is the inductive limit of the family $(A_{λ}^{'})$ of its sub- $A$ -algebras of finite type, ordered by inclusion; if $X_{λ}^{'} = Spec (A_{λ}^{'})$ , $f$ factors as $X^{'} \to X_{λ}^{'} f_{λ} X$ and it results from (1.9.2, (ii)) that one has $f (X^{'}) = ⋂_{λ} f_{λ} (X_{λ}^{'})$ ; if one shows that $f_{λ} (X_{λ}^{'})$ is an intersection of constructible parts of $X$ , the same will be true of $f (X^{'})$ . One may therefore restrict to the case where $A^{'}$ is an $A$ -algebra of finite type. Now, one has the following lemma:

Lemma (1.9.3.1).

Let $A$ be a ring, $A^{'}$ an $A$ -algebra of finite type. Then $A^{'}$ is $A$ -isomorphic to a filtered inductive limit $lim A_{μ}^{''}$ where the $A_{μ}^{''}$ are $A$ -algebras of finite presentation (1.4.1).

Indeed, one may write $A^{'} = B / J$ , with $B = A [T_{1}, \dots, T_{n}]$ and $J$ an ideal of $B$ . But $J$ is the inductive limit of the ideals of finite type $J_{μ}^{'}$ of $B$ , contained in $J$ , ordered by inclusion; as in the category of $B$ -modules the functor $lim$ is exact, one concludes that $A^{'} ≅ lim B / J_{μ}^{'}$ up to an $A$ -isomorphism.

The same reasoning as above then allows one to reduce to the case where $A^{'}$ is an $A$ -algebra of finite presentation; but then $f (X^{'})$ is constructible in $X$ by virtue of Chevalley's theorem (1.8.5), which proves b).

Let us finally prove that b) entails a''). If $E$ is an intersection of a family $(G_{λ})_{λ \in L}$ of constructible parts of $X$ , each $E_{α}$ is the intersection of the $G_{λ} \cap X_{α}$ , which are constructible in $X_{α}$ $(0_{III}, 9.1.8, (i))$ , so one is reduced to the case where $X = Spec (A)$ is affine.

One then knows (1.8.3) that for every $λ \in L$ there exists a morphism $f_{λ} : X_{λ}^{'} \to X$ , where $X_{λ}^{'} = Spec (A_{λ}^{'})$ is affine, such that $f_{λ} (X_{λ}^{'}) = G_{λ}$ . It therefore suffices to prove the following lemma:

Lemma (1.9.3.2).

Let $(A_{λ}^{'})_{λ \in L}$ be a family of $A$ -algebras, $X = Spec (A)$ , $X_{λ}^{'} = Spec (A_{λ}^{'})$ , and let $f_{λ} : X_{λ}^{'} \to X$ be the structure morphism. For every finite subset $J$ of $L$ , set $A_{J}^{'} = ⨂_{λ \in J} A_{λ}^{'}$ (tensor product of $A$ -algebras), $X_{J}^{'} = Spec (A_{J}^{'})$ , and let $f_{J} : X_{J}^{'} \to X$ be the structure morphism. One then has

  f_J(X'_J) = ⋂_{λ ∈ J} f_λ(X'_λ).

If $A^{'} = lim A_{J}^{'}$ , $J$ running over the filtered set of finite parts of $L$ , $X^{'} = Spec (A^{'})$ , and if $f : X^{'} \to X$ is the structure morphism, one has

  f(X') = ⋂_{λ ∈ L} f_λ(X'_λ).

The first assertion results from (I, 3.4.7); the second results from (1.9.2, (ii)), which gives the relation $f (X^{'}) = ⋂_{J} f_{J} (X_{J}^{'})$ .

Definition (1.9.4).

Let $X$ be a topological space. We say that a part $E$ of $X$ is pro-constructible (resp. ind-constructible) in $X$ if, for every $x \in X$ , there exists an open neighbourhood $U$ of $x$ in $X$ such that $E \cap U$ is the intersection (resp. union) of locally constructible parts of $U$ .

Taking account of (1.2.7) and $(0_{III}, 9.1.8 an d 9.1.12)$ , the equivalent conditions of proposition (1.9.3) are therefore expressed by saying that $E$ is pro-constructible in $X$ , the locally constructible sets in $X$ being identical to the constructible sets in $X$ under the hypotheses of (1.9.3).

Proposition (1.9.5).

Let $X$ be a prescheme.

(i) For a part $E$ of $X$ to be pro-constructible, it is necessary and sufficient that $X - E$ be ind-constructible.

(ii) Every finite union and every intersection of pro-constructible parts of $X$ is pro-constructible. Every finite intersection and every union of ind-constructible parts of $X$ is ind-constructible.

(iii) Every intersection (resp. union) of locally constructible parts of $X$ is pro-constructible (resp. ind-constructible). Conversely, if $X$ is quasi-compact and quasi-separated, every pro-constructible (resp. ind-constructible) part of $X$ is an intersection (resp. union) of constructible parts of $X$ .

(iv) Let $(U_{α})$ be an open cover of $X$ . For a part $E$ of $X$ to be pro-constructible (resp. ind-constructible) in $X$ , it is necessary and sufficient that for every $α$ , $E \cap U_{α}$ be pro-constructible (resp. ind-constructible) in $U_{α}$ .

(v) Every pro-constructible part $E$ of $X$ is retrocompact in $X$ . For a locally closed part $E$ of $X$ to be retrocompact in $X$ , it is necessary and sufficient that it be pro-constructible in $X$ .

(vi) Let $f : X^{'} \to X$ be a morphism of preschemes; for every pro-constructible (resp. ind-constructible) part $E$ of $X$ , $f^{- 1} (E)$ is pro-constructible (resp. ind-constructible) in $X^{'}$ .

(vii) Let $f : X^{'} \to X$ be a quasi-compact morphism; for every pro-constructible part $E^{'}$ of $X^{'}$ , $f (E^{'})$ is pro-constructible in $X$ ; in particular $f (X^{'})$ is pro-constructible in $X$ .

(viii) Let $f : X^{'} \to X$ be a morphism locally of finite presentation; for every ind-constructible part $E^{'}$ of $X^{'}$ , $f (E^{'})$ is ind-constructible in $X$ ; in particular $f (X^{'})$ is ind-constructible in $X$ .

(ix) Suppose $X$ is quasi-compact and quasi-separated; then, for a part $E$ of $X$ to be pro-constructible, it is necessary and sufficient that there exist an affine scheme $X^{'}$ and a morphism (necessarily quasi-compact) $f : X^{'} \to X$ such that $E = f (X^{'})$ .

Properties (i), (ii), (iv), and the first assertion of (iii) result from definition (1.9.4) and are valid for an arbitrary topological space, using the mutual distributivity of intersection and union and the fact that if $T$ is locally constructible in $X$ , $T \cap U$ is locally constructible in $U$ for every open $U$ of $X$ . If $X$ is quasi-compact and quasi-separated and $E$ is pro-constructible in $X$ , there is a finite cover $(U_{i})$ of $X$ formed of affine open sets such that $E \cap U_{i}$ is the intersection of constructible parts $M_{iλ}$ of $U_{i}$ ( $λ \in L_{i}$ ); the $M_{iλ}$ are also constructible in $X$ by virtue of (1.2.7) and $(0_{III}, 9.1.8, (ii))$ , and $E$ is the intersection of the finite unions $⋃_{i} M_{i, λ (i)}$ (where $λ (i) \in L_{i}$ for every $i$ ), which are constructible parts of $X$ ; this proves the second assertion of (iii). Assertion (ix) results from (iii) and from (1.9.3). To prove the first assertion of (v), one may restrict to the case where $X$ is affine, and prove then that $E$ is quasi-compact $(0_{III}, 9.1.1)$ ; but since $X$ is then quasi-separated, there exists by virtue of (ix) a quasi-compact morphism $f : X^{'} \to X$ such that $E = f (X^{'})$ ; as $X^{'}$ is quasi-compact, the same is true of its image $E$ under a continuous map.

To prove (vi), one may restrict, by virtue of (iv), to the case where $X$ is affine; the conclusion then results from (iii) and from (1.8.2).

Similarly, to prove (vii), one may restrict to the case where $X$ is affine; then $X^{'}$ is a finite union of affine open sets $X_{i}^{'}$ and $f (E^{'})$ is the union of the $f (E^{'} \cap X_{i}^{'})$ , so that one may also suppose $X^{'}$ affine, by virtue of (ii); but then $E^{'} = g (X^{''})$ , where $g : X^{''} \to X^{'}$ is a quasi-compact morphism, by virtue of (ix), and one has $f (E^{'}) = f (g (X^{''}))$ , hence $f (E^{'})$ is pro-constructible since $g \circ f$ is quasi-compact.

One deduces from (vii) the proof of the second assertion of (v): indeed, let $E$ be a locally closed and retrocompact part of $X$ , and consider a subprescheme of $X$ having $E$ as underlying space (I, 5.2.1); the canonical injection $j : E \to X$ is then by hypothesis a quasi-compact morphism, and it results from (vii) that $E = j (E)$ is pro-constructible in $X$ .

Finally, to prove (viii), one may again suppose $X$ affine; in addition, if $(X_{α}^{'})$ is an affine open cover of $X^{'}$ such that $E^{'} \cap X_{α}^{'}$ is a union of constructible parts of $X_{α}^{'}$ ((iii) and (iv)), $f (E^{'})$ is the union of the $f (E^{'} \cap X_{α}^{'})$ , and, by virtue of Chevalley's theorem (1.8.4), each of the $f (E^{'} \cap X_{α}^{'})$ is a union of constructible parts of $X$ , whence the conclusion.

Exemples (1.9.6).

Every finite part of a prescheme $X$ is pro-constructible: indeed, it suffices to consider the parts $x$ reduced to a single point (1.9.5, (ii)), and $x$ is the image of $Spec (k (x))$ under the canonical morphism $Spec (k (x)) \to X$ , which is quasi-compact; whence the conclusion by (1.9.5, (vii)). By contrast, a part $x$ reduced to a point is not necessarily ind-constructible; for example, let $A$ be a Noetherian integral ring having an infinity of maximal ideals, and let $x$ be the generic point of $X = Spec (A)$ ; if $x$ were ind-constructible in $X$ , it would be constructible by virtue of (1.9.5, (iii)) and consequently would contain a non-empty open of $X$ $(0_{III}, 9.2.2)$ ; but this contradicts the hypothesis that $A$ has infinitely many maximal ideals, by virtue of Artin–Tate's theorem $(0_{I V}, 16.3.3)$ .

Every closed part $Y$ of a prescheme $X$ is pro-constructible, by (1.9.5, (v)). Every open part of $X$ is therefore ind-constructible, but an open part $U$ of $X$ is pro-constructible only if it is retrocompact, again by virtue of (1.9.5, (v)).

Finally, if $X$ is a Noetherian prescheme, hence quasi-separated (1.2.8), it follows from (1.9.5, (iii)) and from $(0_{III}, 9.1.7)$ that the ind-constructible parts of $X$ are the unions of locally closed parts of $X$ .

Theorem (1.9.7).

Let $X$ be a quasi-compact prescheme, $E$ an ind-constructible part of $X$ , $(F_{λ})_{λ \in L}$ a family of pro- constructible parts of $X$ such that $⋂_{λ \in L} F_{λ} \subset E$ ; then there exists a finite subset $J$ of $L$ such that $⋂_{λ \in J} F_{λ} \subset E$ .

Note that the sets $F = X - E$ and $F \cap F_{λ}$ are pro-constructible, so one is reduced to:

Corollary (1.9.8).

Let $X$ be a quasi-compact prescheme, $(F_{λ})_{λ \in L}$ a family of pro-constructible parts of $X$ whose intersection is empty; then there exists a finite subfamily $(F_{λ})_{λ \in J}$ whose intersection is empty.

Covering $X$ by a finite number of affine open sets, and using (1.9.5, (iv)), one is reduced to the case where $X$ is affine; by virtue of (1.9.5, (ix)), one has $F_{λ} = f_{λ} (X_{λ}^{'})$ ,

where $X_{λ}^{'} = Spec (A_{λ}^{'})$ is affine, and $f_{λ}$ is a morphism $X_{λ}^{'} \to X$ ; then, with the notations of (1.9.3.2), one has by hypothesis $f (X^{'}) = \emptyset$ , hence $X^{'} = \emptyset$ ; but this entails by (1.9.2, (i)) that there exists a finite subset $J$ of $L$ such that $X_{J}^{'} = \emptyset$ , hence $f_{J} (X_{J}^{'}) = ⋂_{λ \in J} F_{λ} = \emptyset$ .

Corollary (1.9.9).

Let $X$ be a quasi-compact prescheme, $F$ a pro-constructible part of $X$ , $(E_{λ})_{λ \in L}$ a family of ind- constructible parts of $X$ such that $F \subset ⋃_{λ \in L} E_{λ}$ ; then there exists a finite subset $J$ of $L$ such that $F \subset ⋃_{λ \in J} E_{λ}$ . In particular, from every cover of $X$ by ind-constructible parts, one can extract a finite cover of $X$ .

This results from (1.9.7) by passing to complements.

Proposition (1.9.10).

Let $X$ be a prescheme. For a part $E$ of $X$ to be ind-constructible, it is necessary that, for every $x \in X$ , the intersection $E \cap \overline{x}$ be a neighbourhood of $x$ in $\overline{x}$ . If $X$ is locally Noetherian, this condition is sufficient.

Suppose $E$ is ind-constructible, and let $Y$ be the intersection of $\overline{x}$ and an affine open in $X$ containing $x$ ; there is therefore a subprescheme of $X$ having $Y$ as underlying space (I, 5.2.1), and if $j : Y \to X$ is the canonical injection, $E \cap Y = j^{- 1} (E)$ is ind-constructible in $Y$ (1.9.5, (vi)). Since $Y$ is quasi-compact and separated, $E \cap Y$ is therefore a union of constructible parts of $Y$ (1.9.5, (iii)); consequently, there are two opens $U$ , $V$ in $Y$ such that $x \in U \cap ∁ V \subset E \cap Y$ $(0_{III}, 9.1.3)$ . But as $x$ is generic point of the irreducible space $Y$ , $V$ is necessarily empty and one has $U \subset E \cap Y$ .

Suppose now $X$ locally Noetherian, and let $E$ be a part of $X$ satisfying the condition of the statement. By virtue of definition (1.9.4), one may restrict to the case where $X$ is Noetherian. Then, for every $x \in X$ , $E \cap \overline{x}$ contains a non-empty part of the form $U \cap \overline{x}$ , where $U$ is open in $X$ ; this shows that $E$ is a union of locally closed parts of $X$ , hence is ind-constructible (1.9.6).

Proposition (1.9.11).

Let $X$ be a prescheme, $E$ a part of $X$ . The following properties are equivalent:

a) $E$ is locally constructible.

b) $E$ is ind-constructible and pro-constructible.

c) $E$ and $X - E$ are pro-constructible.

d) $E$ and $X - E$ are ind-constructible.

It manifestly suffices to prove that d) entails a), and one may restrict to the case where $X$ is affine. Then (1.9.5, (iii)), $E$ (resp. $X - E$ ) is a union of a family $(E_{λ})$ (resp. $(E_{μ}^{'})$ ) of constructible parts of $X$ ; as the $E_{λ}$ and the $E_{μ}^{'}$ form a cover of $X$ , it follows from (1.9.9) that there are indices in finite number $λ_{i}$ , $μ_{j}$ such that the $E_{λ_{i}}$ and the $E_{μ_{j}}^{'}$ form a cover of $X$ ; this implies that $E$ is the union of the $E_{λ_{i}}$ , hence is constructible.

Corollary (1.9.12).

Let $f : X \to Y$ be a surjective morphism, which is either quasi-compact or locally of finite presentation. For a part $E$ of $Y$ to be locally constructible (resp. pro-constructible, resp. ind-constructible), it is necessary and sufficient that $f^{- 1} (E)$ be so in $X$ .

One knows that the condition is necessary ((1.8.2) and (1.9.5, (vi))); to show that it is sufficient, one is reduced to the case where $X$ is affine, by virtue of (1.9.5, (iv)); moreover, by virtue of (1.9.11), one may restrict to the case where $f^{- 1} (E)$ is pro-constructible, or to the case where $f^{- 1} (E)$ is ind-constructible. If $f$ is surjective and quasi-compact, and $f^{- 1} (E)$ pro-constructible, it follows from (1.9.5, (vii)) that $E = f (f^{- 1} (E))$ is pro-constructible. If $f$ is surjective and locally of finite presentation, and $f^{- 1} (E)$ ind-constructible, $E = f (f^{- 1} (E))$ is ind-constructible by (1.9.5, (viii)), which completes the proof.

(1.9.13)

Let $X$ be a prescheme, $T$ its topology; it follows from (1.9.5, (i) and (ii)) that the ind-constructible (resp. pro-constructible) parts of $X$ are the open (resp. closed) parts for a topology on $X$ , called the constructible topology and which we shall denote $T_{co n s}$ ; we shall denote by $X^{co n s}$ the set $X$ endowed with the topology $T_{co n s}$ .

Proposition (1.9.14).

Let $X$ be a prescheme, $T$ its topology, $T_{co n s}$ the constructible topology on $X$ .

(i) The topology $T_{co n s}$ is finer than $T$ .

(ii) The locally constructible parts of $X$ are identical to the parts of the space $X^{co n s}$ that are at once open and closed.

(iii) For every morphism $f : X \to Y$ , the underlying map from $X^{co n s}$ to $Y^{co n s}$ is continuous; we denote it $f^{co n s}$ .

(iv) If the morphism $f : X \to Y$ is quasi-compact, $f^{co n s}$ is a closed map; in particular, if $f$ is quasi-compact and bijective, $f^{co n s}$ is a homeomorphism.

(v) If a morphism $f : X \to Y$ is locally of finite presentation, $f^{co n s}$ is an open map; in particular, if $f$ is bijective and locally of finite presentation, $f^{co n s}$ is a homeomorphism.

(vi) For every open $U$ of $X$ , the topology induced by $T_{co n s}$ on $U$ is identical to the topology of $U^{co n s}$ .

Indeed, (i) results from the fact that every open for $T$ is an open for $T_{co n s}$ (1.9.6), and (ii) is a translation of (1.9.11); (iii), (iv), and (v) translate respectively (1.9.5), (vi), (vii), and (viii). Finally to prove (vi), it suffices to remark that the canonical injection $j : U \to X$ is locally of finite presentation (1.4.3), so j^cons : U^cons → X^cons is a continuous open map.

Proposition (1.9.15).

Let $X$ be a prescheme.

(i) For $X^{co n s}$ to be quasi-compact, it is necessary and sufficient that $X$ be quasi-compact.

(ii) For $X^{co n s}$ to be separated, it is necessary and sufficient that $X$ be quasi-separated, and $X^{co n s}$ is then locally compact and totally disconnected.

(iii) For $X^{co n s}$ to be compact, it is necessary and sufficient that $X$ be quasi-compact and quasi-separated.

(iv) Every point of $X$ admits, for the topology $T_{co n s}$ , an open and compact neighbourhood.

(v) For a morphism $f : X \to Y$ to be quasi-compact, it is necessary and sufficient that the continuous map f^cons : X^cons → Y^cons be proper.

(i) Since $T_{co n s}$ is finer than $T$ , it is clear that if $X^{co n s}$ is quasi-compact, so is $X$ ; the converse results from (1.9.9).

(ii) Suppose $X$ quasi-separated, and let us show that $X^{co n s}$ is totally disconnected: indeed, if $x$ , $y$ are two distinct points of $X$ and if for example $x \in / \overline{y}$ , there exists an

affine open neighbourhood $U$ of $x$ in $X$ not containing $y$ ; as $X$ is quasi-separated, $U$ is retrocompact in $X$ (1.2.7), so $U$ and $X - U$ are locally constructible in $X$ , and consequently open in $X^{co n s}$ by virtue of (1.9.11), whence our assertion. Since on every affine open $U$ of $X$ the topology induced by $T_{co n s}$ is that of $U^{co n s}$ by virtue of (1.9.14, (vi)), it follows from the foregoing that $X^{co n s}$ is locally compact, since $U$ is open in $X^{co n s}$ and $U^{co n s}$ is compact. It remains to prove that if $X^{co n s}$ is separated, then $X$ is quasi-separated; consider in effect an affine open $U$ of $X$ ; the topology induced on $U$ by $T_{co n s}$ is that of $U^{co n s}$ (1.9.14, (vi)), so $U$ is compact for this induced topology, since $U^{co n s}$ is compact by the first part of the reasoning. If $V$ is a second affine open in $X$ , $U \cap V$ is then a compact part of the separated space $X^{co n s}$ , being the intersection of two compact parts of this space; as the topology induced by $T_{co n s}$ on $U \cap V$ is that of $(U \cap V)^{co n s}$ (1.9.14, (vi)) and the identity map $(U \cap V)^{co n s} \to U \cap V$ is continuous, one concludes that $U \cap V$ is quasi-compact for the topology induced by $T$ ; $X$ is consequently quasi-separated by virtue of (1.2.7).

(iii) is an immediate corollary of (i) and (ii).

(iv) For every $x \in X$ , an affine open neighbourhood $U$ of $x$ for $T$ is also a neighbourhood of $x$ for $T_{co n s}$ , and it is compact by virtue of (iii) and (1.9.14, (vi)), the topology induced on $U$ by $T_{co n s}$ being identical to that of $U^{co n s}$ .

(v) Suppose $f$ quasi-compact; then one already knows (1.9.14, (iv)) that $f^{co n s}$ is a closed map. Let on the other hand $y$ be a point of $Y$ , and set

  Z = f⁻¹(y) = X ×_Y Spec(k(y));

$Z$ is quasi-compact (1.1.2, (iii)) and as the canonical morphism $p : Z \to X$ is injective, it results from (i) and from the fact that the map $p^{co n s}$ is continuous that the topology induced on $f^{- 1} (y)$ by that of $X^{co n s}$ makes $f^{- 1} (y)$ a quasi-compact space. This proves that $f^{co n s}$ is a proper map (Bourbaki, Top. gén., chap. I, 3e éd., §10, n° 2, th. 1). Conversely, suppose the continuous map $f^{co n s}$ is proper, and let $V$ be a quasi-compact open of $Y$ ; if $h : V \to Y$ is the canonical injection, h^cons : V^cons → Y^cons is continuous and injective and $V^{co n s}$ is quasi-compact by (i), so the topology induced on $V$ by that of $Y^{co n s}$ makes $V$ a quasi-compact space. The hypothesis that $f^{co n s}$ is proper then entails that the topology induced on $f^{- 1} (V)$ by that of $X^{co n s}$ makes $f^{- 1} (V)$ a quasi-compact space (loc. cit., prop. 6), so $f^{- 1} (V)$ is also a quasi-compact subspace of $X$ , which shows that the morphism $f$ is quasi-compact.

(1.9.16)

We shall show later how one may, for every prescheme $X$ , endow the space $X^{co n s}$ with a sheaf of rings making it a prescheme whose local rings are all fields, identical to the residue fields at the points of $X$ . Such preschemes are introduced, for example, in a natural way in the translation, in the language of schemes, of Néron's constructions [32] in his theory of the reduction of abelian varieties.

1.10. Application to open morphisms

Theorem (1.10.1).

Let $X$ be a prescheme, $E$ an ind-constructible part of $X$ (1.9.4), $x$ a point of $X$ . For $x$ to be interior to $E$ , it is necessary and sufficient that every generization $(0_{I}, 2.1.2)$ $x^{'}$ of $x$ belong to $E$ , in other words (I, 2.4.2) that one have $Spec (O_{X, x}) \subset E$ .

The condition is obviously necessary, every neighbourhood of $x$ containing the generizations of $x$ . To see that it is sufficient, one may obviously restrict to the case where $X = Spec (A)$ is affine, $x$ being a prime ideal $p$ of $A$ . One then knows (1.9.5, (ix)) that there exists an affine scheme $X^{'} = Spec (A^{'})$ and a morphism $f : X^{'} \to X$ such that $f (X^{'}) = X - E$ . Set $Y = Spec (O_{X, x})$ and $Y^{'} = X^{'} \times_{X} Y$ ; the hypothesis means (by virtue of (I, 3.6.5)) that one has $Y^{'} = \emptyset$ , in other words $A^{'} \otimes_{A} A_{p} = 0$ . As $A_{p} = lim A_{t}$ , where $t$ runs over $A - p$ $(0_{I}, 1.4.5)$ , one has $lim A_{t}^{'} = 0$ , the functor $lim$ commuting with the tensor product. Consequently (1.9.1), there exists a $t \in A - p$ such that $A_{t}^{'} = 0$ , and $D (t)$ is then an open neighbourhood of $x$ in $X$ contained in $E$ .

Corollary (1.10.2).

Let $f : X \to Y$ be a morphism locally of finite presentation, and let $y \in Y$ . For $y$ to be interior to $f (X)$ it is necessary and sufficient that every generization $y^{'}$ of $y$ belong to $f (X)$ .

One knows indeed (1.9.5, (viii)) that $f (X)$ is ind-constructible in $Y$ , and it suffices to apply (1.10.1).

We shall say that a continuous map $f : X \to Y$ is open at a point $x \in X$ if the image under $f$ of every neighbourhood of $x$ in $X$ is a neighbourhood of $f (x)$ in $Y$ .

Proposition (1.10.3).

Let $f : X \to Y$ be a morphism, $x$ a point of $X$ , $y = f (x)$ . Consider the following conditions:

a) $f$ is open at the point $x$ .

b) For every generization $y^{'}$ of $y$ , there exists a generization $x^{'}$ of $x$ such that $f (x^{'}) = y^{'}$ .

b') The image under $f$ of $Spec (O_{X, x})$ is $Spec (O_{Y, y})$ .

c) For every irreducible closed part $Y^{'}$ of $Y$ containing $y$ , there exists an irreducible component of $X^{'} = f^{- 1} (Y^{'})$ containing $x$ and dominating $Y^{'}$ .

Then one has the implications a) ⟹ b) ⟺ b') ⟸ c). If in addition $f$ is locally of finite presentation, the four conditions are equivalent.

By definition of a generization, the image under $f$ of $Spec (O_{X, x})$ is always contained in $Spec (O_{Y, y})$ , so b) and b') are equivalent. It is immediate that c) entails b), because if $y^{'}$ is a generization of $y$ , $Y^{'} = \overline{y^{'}}$ is an irreducible part of $Y$ containing $y$ ; if $x^{'}$ is the generic point of an irreducible component of $f^{- 1} (Y^{'})$ which contains $x$ and dominates $Y^{'}$ , one has $f (x^{'}) = y^{'}$ $(0_{I}, 2.1.5)$ and $x^{'}$ is a generization of $x$ . Conversely, b) entails c): indeed, let $y^{'}$ be the generic point of $Y^{'}$ and let $x^{'}$ be a generization of $x$ such that $f (x^{'}) = y^{'}$ ; let $Z^{'}$ be an irreducible component of $f^{- 1} (Y^{'})$ containing $x^{'}$ (hence $x$ ); as $f (x^{'})$ is already dense in $Y^{'}$ , a fortiori so is $f (Z^{'})$ .

To see that a) entails b'), one may obviously restrict to the case where $X = Spec (B)$ and $Y = Spec (A)$ are affine, $x$ being a prime ideal $q$ of $B$ , so that $O_{X, x} = B_{q} = lim B_{t}$ ,

where $t$ runs over $B - q$ . One then has, by (1.9.2), $f (Spec (O_{X, x})) = ⋂_{t} f (Spec (B_{t})) = ⋂_{t} f (D (t))$ , and by hypothesis, for every $t \in B - q$ , $y$ is interior to $f (D (t))$ , so $f (D (t))$ contains $Spec (O_{Y, y})$ ; whence $Spec (O_{Y, y}) \subset f (Spec (O_{X, x}))$ , which establishes our assertion. Finally, when $f$ is locally of finite presentation, b) implies a) by virtue of (1.10.2) applied to the restriction $U \to Y$ of $f$ to an arbitrary open neighbourhood $U$ of $x$ .

Corollary (1.10.4).

Let $f : X \to Y$ be a morphism. Consider the following conditions:

a) $f$ is open.

b) For every $x \in X$ and every generization $y^{'}$ of $y = f (x)$ , there exists a generization $x^{'}$ of $x$ such that $f (x^{'}) = y^{'}$ .

b') For every $x \in X$ , the image under $f$ of $Spec (O_{X, x})$ is $Spec (O_{Y, f (x)})$ .

c) For every irreducible closed part $Y^{'}$ of $Y$ , every irreducible component of $f^{- 1} (Y^{'})$ dominates $Y^{'}$ .

Then one has the implications a) ⟹ b) ⟺ b') ⟸ c). If in addition $f$ is locally of finite presentation, the four conditions are equivalent.

To say that $f$ is open means that $f$ is open at every point $x \in X$ , so the implications a) ⟹ b) ⟺ b') result from the analogous implications in (1.10.3), as does the implication b) ⟹ a) when $f$ is locally of finite presentation. The implication c) ⟹ b) also results from the analogous implication in (1.10.3); let us finally prove that b) entails c). Indeed, let $x^{'}$ be the generic point of an irreducible component of $f^{- 1} (Y^{'})$ , and let us show that $y^{'} = f (x^{'})$ is the generic point of $Y^{'}$ . Let y'' be a generization of $y^{'}$ ; there exists by hypothesis a generization x'' of $x^{'}$ such that $f (x^{''}) = y^{''}$ , and as $x^{''} \in f^{- 1} (Y^{'})$ , one has necessarily $x^{''} = x^{'}$ , hence $y^{''} = y^{'}$ , which completes the proof.

(To be continued.)

Éléments de Géométrie Algébrique — English