Éléments de Géométrie Algébrique — English

§9. Constructible properties

Let $S$ be a prescheme, $f:X\to S$ a morphism of finite presentation (1.6.1), $\mathcal{F}$ a quasi-coherent ${\mathcal{O}}_X$-Module of finite presentation. We propose, in this section, to give criteria ensuring, for example, that the set of $s\in S$ such that the $k(s)$-prescheme $X_s=f^{-1}(s)=X{\times }_S\mathrm{Spec}(k(s))$ has a certain property, or such that the ${\mathcal{O}}_{X_s}$-Module ${\mathcal{F}}_s=\mathcal{F}{\otimes }_{{\mathcal{O}}_S}k(s)$ has a certain property, is locally constructible, or at least ind-constructible (1.9.4); we shall see that this is the case for most of the properties that arise in algebraic geometry. Assuming only $f$ locally of finite presentation, we shall also give (9.9) criteria for the set of points $x\in X$ where the fibre $X_{f(x)}$ (or the ${\mathcal{O}}_{X_{f(x)}}$-Module ${\mathcal{F}}_{f(x)}$) has a certain property to be locally constructible. We shall see in §12 that these results, combined with the additional hypothesis that $f$ is flat (resp. proper and flat), allow one to prove that the sets considered in $X$ (resp. in $S$) are even open in many cases.

9.1. The principle of finite extension

Proposition (9.1.1) (Principle of finite extension).

Let $k$ be a field, $\mathcal{C}$ a set of extensions of $k$. Assume the following conditions are satisfied:

(i) If $K\in \mathcal{C}$ and there exists a $k$-homomorphism $K\to K^{\prime }$ (where $K^{\prime }$ belongs to the universe in which one places oneself), then $K^{\prime }\in \mathcal{C}$.

(ii) If $K\in \mathcal{C}$, there exists a sub-extension $K^{\prime }$ of $K$, of finite type over $k$, such that $K^{\prime }\in \mathcal{C}$.

(iii) If $K\in \mathcal{C}$ is the field of fractions of an integral $k$-algebra $A$ of finite type, there exists $f\in A-0$ such that for every maximal ideal $\mathfrak{m}$ of $A_f$ one has $A_f/\mathfrak{m}\in \mathcal{C}$.

Let $\Omega$ be an algebraically closed extension of $k$ (in the universe under consideration). The following conditions are equivalent:

a) $\mathcal{C}$ is non-empty.

b) There exists $K^{\prime }\in \mathcal{C}$ which is a finite extension of $k$.

c) One has $\Omega \in \mathcal{C}$.

Condition (i) evidently implies that b) entails c), and c) entails trivially a); let us prove that a) entails b). By virtue of (ii) and (iii) there exists an extension $K^{\prime }\in \mathcal{C}$ of the form $A/\mathfrak{m}$, where $A$ is a $k$-algebra of finite type over $k$ and $\mathfrak{m}$ is a maximal ideal of $A$. One knows, by Hilbert's Nullstellensatz (Bourbaki, Alg. comm., chap. V, §3, n° 1, cor. 2 of th. 1), that $K^{\prime }$ is a finite extension of $k$.

Corollary (9.1.2).

Under the hypotheses (i), (ii), (iii) of (9.1.1), if $k$ is algebraically closed and if $\mathcal{C}$ is non-empty, then $\mathcal{C}$ contains all extensions of $k$ in the universe considered.

Indeed, since a) entails c), one has $k\in \mathcal{C}$, and the conclusion results from hypothesis (i).

Remark (9.1.3).

In practice, one will verify condition (ii) of (9.1.1) by noting that $K$ is the inductive limit of its sub-extensions $K_{\alpha }$ of finite type, and by applying the results of §8, taking into account if necessary that for $K_{\alpha }\subset K_{\beta }$, $K_{\beta }$ is faithfully flat over $K_{\alpha }$. Frequently the set $\mathcal{C}$ is formed of the fields belonging to a set ${\mathcal{C}}^{\prime }$ of $k$-algebras that satisfies the following condition:

(i bis) If $A\in {\mathcal{C}}^{\prime }$ and there exists a $k$-algebra homomorphism $A\to A^{\prime }$ (where $A^{\prime }$ belongs to the universe in which one places oneself), then $A^{\prime }\in {\mathcal{C}}^{\prime }$.

When this is the case, condition (i) is trivially satisfied, and one will generally verify condition (iii) of (9.1.1) by noting that the field of fractions $K$ of $A$ is the inductive limit of the algebras $A_f$ (for the relation $D(f)\supset D(g)$), and by applying the results of §8, taking into account if necessary that the morphisms $D(g)\to D(f)$ are open immersions.

By contrast, when (i bis) is not satisfied, the proof of (iii) is often more delicate, and is tied to constructibility criteria that will be developed below.

Here are some typical examples of application of the principle of finite extension.

Proposition (9.1.4).

Let $k$ be a field, $\Omega$ an algebraically closed extension of $k$, $X$ and $Y$ two preschemes of finite type over $k$. The following conditions are equivalent:

a) There exists an $\Omega$-morphism $X_{(\Omega )}\to Y_{(\Omega )}$ (resp. an $\Omega$-morphism possessing one (specified) of the properties (i) to (xiv) of (8.10.5)).

b) There exists a finite extension $k^{\prime }$ of $k$ and a $k^{\prime }$-morphism $X_{(k^{\prime })}\to Y_{(k^{\prime })}$ (resp. a $k^{\prime }$-morphism possessing the property considered).

c) There exists an extension $K$ of $k$ and a $K$-morphism $X_{(K)}\to Y_{(K)}$ (resp. a $K$-morphism possessing the property considered).

One applies remark (9.1.3), taking for ${\mathcal{C}}^{\prime }$ the set of all $k$-algebras $A$ (of the universe in which one is placed) such that there exists an $A$-morphism $X{\otimes }_{k}A\to Y{\otimes }_{k}A$ (resp. a morphism having that one of the properties of (8.10.5) that one considers). Condition (i bis) of (9.1.3) is then verified thanks to the fact that the

properties envisaged in (8.10.5) are all stable under base change. Condition (iii) of (9.1.1) is therefore satisfied by virtue of (8.8.2, (i)) (resp. (8.10.5)), since $\mathrm{Spec}(k)$ is quasi-compact and quasi-separated. It remains to verify condition (ii) of (9.1.1), which follows again from (8.8.2, (i)) and (8.10.5), by viewing $K$ as the inductive limit of its sub-extensions of finite type. One concludes therefore by (9.1.1).

In particular, if there exists an extension $K$ of $k$ and a $K$-isomorphism $X_{(K)}\stackrel{\sim }{\to }{Y}_{(K)}$, one says that $X$ and $Y$ are geometrically isomorphic.

The following corollary generalizes (II, 6.6.5).

Corollary (9.1.5).

Let $k$ be a field, $X$ a $k$-prescheme. If there exists an extension $K$ of $k$ such that $X_{(K)}$ is projective (resp. quasi-projective) over $K$, then $X$ is projective (resp. quasi-projective) over $k$.

The morphism $\mathrm{Spec}(K)\to \mathrm{Spec}(k)$ being faithfully flat and quasi-compact, it follows already from (2.7.1, (v)) that $X$ is of finite type over $k$. The hypothesis means that there exists a closed immersion (resp. an immersion) $X_{(K)}\to {\mathbf{P}}_K^r={\mathbf{P}}_k^r{\otimes }_{k}K$ (II, 5.5.4, (ii) and 5.5.3); applying (9.1.4) for property (iv) (resp. (ii)) of (8.10.5), one deduces that there is a finite extension $k^{\prime }$ of $k$ and a closed immersion (resp. an immersion) $X_{(k^{\prime })}\to {\mathbf{P}}_{k^{\prime }}^r$, that is, $X_{(k^{\prime })}$ is projective (resp. quasi-projective) over $k^{\prime }$. One concludes by (II, 6.6.5).

Proposition (9.1.6).

Let $k$ be a field, $\Omega$ an algebraically closed extension of $k$, $X$ a prescheme of finite type over $k$, $\mathcal{F}$, $\mathcal{G}$ two coherent ${\mathcal{O}}_X$-Modules. Suppose there exists an isomorphism $\mathcal{F}{\otimes }_k\Omega \stackrel{\sim }{\to }\mathcal{G}{\otimes }_k\Omega$. Then there exist a finite extension $k^{\prime }$ of $k$ and an isomorphism $\mathcal{F}{\otimes }_k{k}^{\prime }\stackrel{\sim }{\to }\mathcal{G}{\otimes }_k{k}^{\prime }$.

The reasoning is the same as in (9.1.4), applying (8.5.2, (i)) (one uses here, in the proof of property (iii) of (9.1.1), the fact that the morphisms $D(g)\to D(f)$ (with the notation of (9.1.3)) are open immersions, and a fortiori flat morphisms).

9.2. Constructible and ind-constructible properties

Definition (9.2.1).

Let $P(X,\mathcal{F},k)$ be a relation. We say that $P$ is a constructible (resp. ind-constructible) property of algebraic preschemes if the following two conditions are satisfied:

(i) If $k$ is a field, $X$ an algebraic prescheme over $k$, $\mathcal{F}$ a coherent ${\mathcal{O}}_X$-Module, $k^{\prime }$ an extension of $k$, then, for $P(X,\mathcal{F},k)$ to be true, it is necessary and sufficient that $P(X_{(k^{\prime })},\mathcal{F}{\otimes }_k{k}^{\prime },k^{\prime })$ be true.

(ii) Let $S$ be an integral Noetherian prescheme, of generic point $\eta$, $u:X\to S$ a morphism of finite type, $\mathcal{F}$ a coherent ${\mathcal{O}}_X$-Module. For every $s\in S$, set $X_s=u^{-1}(s)=X{\times }_S\mathrm{Spec}(k(s))$, ${\mathcal{F}}_s=\mathcal{F}{\otimes }_{{\mathcal{O}}_S}k(s)$. Let $E$ be the set of $s\in S$ such that $P(X_s,{\mathcal{F}}_s,k(s))$ is true. Then one of the sets $E$, $S-E$ (resp. the set $E$) contains a non-empty open set (and consequently is a neighbourhood of $\eta$) (resp. contains a non-empty open set if it contains $\eta$).

Remarks (9.2.2).

(i) This is of course a convention of language of metamathematical nature and not a mathematical definition properly speaking. One has analogous "definitions" for relations between $k$, one or several algebraic $k$-preschemes, $k$-morphisms between these preschemes, coherent Modules on these preschemes, or homomorphisms between these Modules.

(ii) We shall also have to consider relations in which constructible parts of preschemes figure. For example, let $P(X,Z,k)$ be a relation; we shall say (by abuse of language) that $P$ is a constructible (resp. ind-constructible) property of the constructible part $Z$ of $X$ if the following two conditions are satisfied:

1° If $k$ is a field, $X$ an algebraic prescheme over $k$, $Z$ a constructible part of $X$, $k^{\prime }$ an extension of $k$, then, for $P(X,Z,k)$ to be true, it is necessary and sufficient that $P(X_{(k^{\prime })},p^{-1}(Z),k^{\prime })$ be true ($p:X_{(k^{\prime })}\to X$ being the canonical projection).

2° Let $S$ be an integral Noetherian prescheme, of generic point $\eta$, $u:X\to S$ a morphism of finite type, $Z$ a constructible part of $X$. For every $s\in S$, set $X_s=u^{-1}(s)$, $Z_s=Z\cap X_s$. Let $E$ be the set of $s\in S$ such that $P(X_s,Z_s,k(s))$ is true. Then one of the sets $E$, $S-E$ (resp. the set $E$) contains a non-empty open set (resp. contains a non-empty open set if it contains $\eta$).

One should note that in condition 2° one must assume that $Z$ is a constructible part of $X$, and not only that $Z_s$ is a constructible part of $X_s$ for every $s$; the former of these two properties entails the latter (1.8.2), but not conversely.

(iii) If $P$ is a constructible property, it is clear that the same is true of "not $P$". If $P$, $Q$ are two constructible (resp. ind-constructible) properties, the same is true of the properties "$P$ or $Q$" and "$P$ and $Q$"; indeed, if, under the hypotheses of (9.2.1, (ii)), $E$, $E^{\prime }$ are two parts of $S$ and if $E$ contains a non-empty open set, the same is true of $E\cup E^{\prime }$, and if $S-E$ and $S-E^{\prime }$ each contain a non-empty open set, the same is true of $S-(E\cup E^{\prime })=(S-E)\cap (S-E^{\prime })$.

(iv) Let $P(X,\mathcal{F},k)$ be a relation satisfying condition (9.2.1, (i)); let $S$ be a prescheme, $u:X\to S$ a morphism of finite type; with the notation of (9.2.1, (ii)), let $E$ be the set of $s\in S$ such that $P(X_s,{\mathcal{F}}_s,k(s))$ is true. Let on the other hand $g:S^{\prime }\to S$ be an arbitrary morphism, and set $X^{\prime }=X_{(S^{\prime })}$, ${\mathcal{F}}^{\prime }=\mathcal{F}{\otimes }_{{\mathcal{O}}_S}{\mathcal{O}}_{S^{\prime }}$; then it follows from the transitivity of fibres (I, 3.6.4) and from condition (9.2.1, (i)) that the set of $s^{\prime }\in S^{\prime }$ such that $P(X_{s^{\prime }}^{\prime },{\mathcal{F}}_{s^{\prime }}^{\prime },k(s^{\prime }))$ is true is equal to $g^{-1}(E)$. This extends at once to the case where several preschemes, Modules, morphisms of preschemes, or homomorphisms of Modules figure in $P$, and to properties of the type considered in (ii).

(v) As we shall see in the remainder of this section and in the course of the rest of Chap. IV, most properties $P$ satisfying condition (9.2.1, (i)) also satisfy (9.2.1, (ii)). As possible exceptions, let us note the property of being projective, or quasi-projective, or affine, or quasi-affine over the base field (for an algebraic prescheme); we shall see (9.6.2) that these properties are ind-constructible, but we shall later give an example where $S$ is a non-empty open part of $\mathrm{Spec}(\mathbb{Z})$ (or an open part of an elliptic curve over a finite field) and where all the fibres $X_s$

except the generic fibre $X_{\eta }$ are projective over $k(s)$ (all the preschemes $X_s$ being of dimension 2).

Proposition (9.2.3).

Let $P$ be a constructible (resp. ind-constructible) property of algebraic preschemes, $S$ a prescheme, $X$ a prescheme of finite presentation over $S$, $\mathcal{F}$ a quasi-coherent ${\mathcal{O}}_X$-Module of finite presentation. Then the set $E$ of $s\in S$ such that $P(X_s,{\mathcal{F}}_s,k(s))$ is true is locally constructible (resp. ind-constructible). Moreover, if $S$ is irreducible of generic point $\eta$, then one of the two sets $E$, $S-E$ is a neighbourhood of $\eta$ in $S$ (resp. $E$ is a neighbourhood of $\eta$ if it contains this point).

To prove these assertions, one may restrict to the case where $S=\mathrm{Spec}(A)$ is affine. One then knows that there exists a sub-ring A_0 of $A$ which is a $\mathbb{Z}$-algebra of finite type, an A_0-prescheme of finite type X_0, and a coherent ${\mathcal{O}}_{X_0}$-Module ${\mathcal{F}}_0$ such that $X$ is isomorphic to $X_0{\otimes }_{A_0}A$ and $\mathcal{F}$ to ${\mathcal{F}}_0{\otimes }_{A_0}A$ (8.9.1). Let $p:S\to S_0=\mathrm{Spec}(A_0)$ be the morphism corresponding to the injection $A_0\to A$, and let E_0 be the set of $s_0\in S_0$ such that $P((X_0{)}_{s_0},({\mathcal{F}}_0{)}_{s_0},k(s_0))$ is true; then, by virtue of (9.2.2, (iv)), one has $E=p^{-1}(E_0)$; one may therefore (1.8.2) restrict to the case where $S$ is the spectrum of a $\mathbb{Z}$-algebra of finite type, hence a Noetherian scheme. Let us use the constructibility criterion $(0_{III},\mathrm{9.2.3})$ (resp. the ind-constructibility criterion (1.9.10)); one is then reduced, using as above (9.2.2, (iv)) and replacing $S$ by an integral closed sub-scheme of $S$, to the case where $S$ is Noetherian and integral, and one must prove that $E$ is rare in $S$ or contains a non-empty open set of $S$ (resp. that $E$ contains a non-empty open set of $S$ if it contains the generic point); but this is guaranteed by virtue of condition (9.2.1, (ii)).

One should note that one has used (9.2.1, (ii)) only when $S$ is the spectrum of an integral ring of finite type over $\mathbb{Z}$. It is clear on the other hand that the statement of (9.2.3) also applies when several preschemes, Modules on these preschemes, morphisms of preschemes, or homomorphisms of Modules figure in $P$. It still applies when (finitely many) parts of the preschemes considered figure in it, provided that one imposes on these parts the condition of being locally constructible. Indeed, the restriction to the case where $S$ is affine shows that one may restrict to the case where these parts are constructible: one then applies (8.3.11), which shows (with the notation above) that a constructible part of $X$ is the inverse image of a constructible part of X_0 for a suitable choice of A_0.

Corollary (9.2.4).

Let $P$ be a constructible (resp. ind-constructible) property of algebraic preschemes, $X$, $Y$ two $S$-preschemes of finite presentation, $f:X\to Y$ an $S$-morphism. For every $s\in S$, set $X_s=X{\times }_S\mathrm{Spec}(k(s))$, $Y_s=Y{\times }_S\mathrm{Spec}(k(s))$, $f_s=f\times 1:X_s\to Y_s$. Then the set $E$ of $s\in S$ such that, for every $y\in Y_s$, the property $P(f_s^{-1}(y),k(y))$ is true, is locally constructible (resp. ind-constructible).

Indeed, let $Z$ be the set of $y\in Y$ such that $P(f^{-1}(y),k(y))$ is true. As the fibres $f^{-1}(y)$ and $f_s^{-1}(y)$ are isomorphic, one sees that $E$ is the set of $s\in S$ such that $Y_s\subset Z$; if $g:Y\to S$ is the structure morphism, one therefore has $E=S-g(Y-Z)$.

Now $f$ is of finite presentation (1.6.2, (v)), so it follows from (9.2.3) that $Z$ is locally constructible (resp. ind-constructible) in $Y$, hence $Y-Z$ is locally constructible (resp. pro-constructible) in $Y$. Since $g$ is of finite presentation, $g(Y-Z)$ is locally constructible (resp. pro-constructible) in $S$, by virtue of Chevalley's theorem (1.8.4) (resp. of (1.9.5, (vii))); hence $E$ is locally constructible (resp. ind-constructible) in $S$.

Remark (9.2.5).

One should note that if $P$ is a property of algebraic preschemes for which prop. (9.2.3) is true, then $P$ also satisfies condition (9.2.1, (ii)): this follows from the fact that in an irreducible Noetherian space, a constructible set is rare or contains a non-empty open set $(0_{III},\mathrm{9.2.3})$.

Proposition (9.2.6).

Let $P$ denote one of the following properties of a $k$-prescheme $X$:

(i) $X$ is empty.

(ii) $X$ is finite over $k$.

(iii) $X$ is radicial over $k$.

(iv) $\dim (X)$ belongs to a given part $\Phi$ of the set $\overline{\mathbb{Z}}=\mathbb{Z}\cup -\infty$.

Then $P$ is constructible.

It is clear that (i) and (ii) are special cases of (iv), taking respectively for $\Phi$ the set $-\infty$ and the set $-\infty ,0$. One has therefore only to prove (iii) and (iv). In each of these two cases condition (i) of (9.2.1) is fulfilled by virtue of (2.7.1, (xv)) and (4.1.4). On the other hand, in case (iii), the property $P$ satisfies the conclusion of (9.2.3) by virtue of (1.8.7); it remains therefore to see that the same is true in case (iv). This will result from the more precise proposition that follows.

Proposition (9.2.6.1).

If $f:X\to S$ is a morphism of finite presentation, the function $s\mapsto \dim (f^{-1}(s))$ is locally constructible.

The question is local on $S$, so one may suppose that $S=\mathrm{Spec}(A)$ is affine and prove that for every $n$, the set $E$ of $s\in S$ such that $\dim (X_s)=n$ is constructible. The same reasoning as in (9.2.3) reduces to the case where $A$ is Noetherian and integral, and it then suffices to prove:

Corollary (9.2.6.2).

Let $S$ be an integral Noetherian prescheme of generic point $\eta$, $f:X\to S$ a morphism of finite type. Then there exists a neighbourhood of $\eta$ in $S$ such that the function $s\mapsto \dim (X_s)$ is constant in this neighbourhood.

The images by $f$ of the irreducible components (finitely many) of $X$ which do not meet $X_{\eta }$ are contained in closed parts of $S$ not containing $\eta$ (since $S$ is integral $(0_I,\mathrm{2.1.5})$), so (replacing $S$ by an open neighbourhood of $\eta$) one may restrict to the case where all the irreducible components $X_i$ of $X$ meet $X_{\eta }$; denote again by $X_i$ the reduced closed sub-prescheme of $X$ having $X_i$ as underlying space; since dim(X_s) = sup_i dim((X_i)_s) (4.1.1), one may restrict

to the case where $X$ is irreducible. There then exists a finite cover $(U_j)$ of $X$ by everywhere-dense affine open sets, and the numbers $\dim ((U_j{)}_{\eta })$ are all equal to $n=\dim (X_{\eta })$ (4.1.1.3); one may therefore restrict to the case where $X$ is affine, hence also $X_{\eta }$. There then exists, by virtue of (4.1.2), a non-empty open set $W$ of $X$ such that $W\subset X_{\eta }$, and a finite surjective $k(\eta )$-morphism $h:W_{\eta }\to {\mathbf{V}}_{k(\eta )}^n(=\mathrm{Spec}(k(\eta )[T_1,\cdots ,T_n]))$; applying (8.8.2, (i)) and the method of (8.1.2, a)), one deduces (replacing $S$ if necessary by a neighbourhood of $\eta$) that $h=g_{\eta }$, where $g:W\to {\mathbf{V}}_A^n(=\mathrm{Spec}(A[T_1,\cdots ,T_n]))$ is an $S$-morphism, and one may suppose this morphism finite and surjective by virtue of (8.10.5, (vi) and (x)). One concludes that for every $s\in S$, the morphism $g_s:W_s\to {\mathbf{V}}_{k(s)}^n$ is finite and surjective, hence $\dim (W_s)=n$ (4.1.2).

9.3. Constructible properties of morphisms of algebraic preschemes

Proposition (9.3.1).

Let $P(X,k)$ be a constructible (resp. ind-constructible) property of algebraic preschemes. Denote by P'(f, X, Y, k) the following relation: $f:X\to Y$ is a $k$-morphism of algebraic $k$-preschemes such that for every $y\in Y$, one has the property $P(f^{-1}(y),k(y))$. Then $P^{\prime }$ is a constructible (resp. ind-constructible) property.

Indeed, since $P$ satisfies condition (9.2.1, (i)), the same is true of $P^{\prime }$ by virtue of the transitivity of fibres (I, 3.6.4); on the other hand, the fact that $P^{\prime }$ satisfies condition (9.2.1, (ii)) results from (9.2.4), in view of remark (9.2.5).

Proposition (9.3.2).

Let $P$ denote one of the following properties of a $k$-morphism $f:X\to Y$ of algebraic $k$-preschemes:

(i) $f$ is surjective.

(ii) $f$ is quasi-finite.

(iii) $f$ is radicial.

(iv) For every $y\in Y$, $\dim (f^{-1}(y))$ belongs to $\Phi$ (notation of (9.2.6)).

Then $P$ is constructible.

This follows at once from (9.3.1) and (9.2.6) if one takes into account that $f$ is of finite type (1.5.4, (v)), the characterization of radicial morphisms (I, 3.5.8), and that of quasi-finite morphisms (II, 6.2.2).

Proposition (9.3.3).

Suppose the hypotheses of (8.8.1) are satisfied, the notation of which we retain; suppose in addition that $S_{\alpha }$ is quasi-compact, $X_{\alpha }$ and $Y_{\alpha }$ of finite presentation over $S_{\alpha }$, and let $f_{\alpha }:X_{\alpha }\to Y_{\alpha }$ be an $S_{\alpha }$-morphism. Let $P$ be an ind-constructible property of morphisms of algebraic preschemes. For every $s\in S$ (resp. $s_{\lambda }\in S_{\lambda }$) set $X_s=X{\times }_S\mathrm{Spec}(k(s))$, $Y_s=Y{\times }_S\mathrm{Spec}(k(s))$, $f_s=f\times 1:X_s\to Y_s$ (resp. $X_{\lambda ,s_{\lambda }}=X_{\lambda }{\times }_{S_{\lambda }}\mathrm{Spec}(k(s_{\lambda }))$, $Y_{\lambda ,s_{\lambda }}=Y_{\lambda }{\times }_{S_{\lambda }}\mathrm{Spec}(k(s_{\lambda }))$, $f_{\lambda ,s_{\lambda }}=f_{\lambda }\times 1:X_{\lambda ,s_{\lambda }}\to Y_{\lambda ,s_{\lambda }}$). Then, in order that for every $s\in S$ one has the property $P(f_s)$, it is necessary and sufficient that there exist $\lambda ⩾\alpha$ such that for every $s_{\lambda }\in S_{\lambda }$, one has $P(f_{\lambda ,s_{\lambda }})$.

Indeed, let $E$ (resp. $E_{\lambda }$) be the set of $s\in S$ (resp. $s_{\lambda }\in S_{\lambda }$) such that the property $P(f_s)$ (resp. $P(f_{\lambda ,s_{\lambda }})$) is true; it follows from (9.2.2, (iv)) that one has $E_{\mu }=u_{\lambda \mu }^{-1}(E_{\lambda })$ for $\lambda ⩽\mu$, and $E=u_{\lambda }^{-1}(E_{\lambda })$; moreover, by virtue of (9.2.3), $E$ (resp. $E_{\lambda }$) is ind-constructible in $S$ (resp. $S_{\lambda }$); the proposition therefore results from (8.3.4) applied to the ind-constructible part $E$ of $S$.

This result generalizes without difficulty to properties $P$ of the type considered in (9.2.3, (i) and (ii)).

Remark (9.3.4).

The conjunction of (9.3.3) and (9.3.2, (ii)) proves the assertion (8.10.5, (xi)).

Proposition (9.3.5).

Let $P(X,Y,k)$ be the property: "$X$ and $Y$ are two preschemes of finite type over the field $k$, and there exists an extension $k^{\prime }$ of $k$ and a $k^{\prime }$-morphism $g:X_{(k^{\prime })}\to Y_{(k^{\prime })}$ satisfying $Q(g)$", where $Q$ is one of the properties (i) to (xiv) of (8.10.5). Then $P$ is an ind-constructible property.

The definition of $P$ shows indeed that condition (9.2.1, (i)) is satisfied, taking account of the fact that the property $Q(g)$ is stable under change of base field, and that two extensions of $k$ can always be considered as sub-extensions of a third extension of $k$. To verify (9.2.1, (ii)), one may restrict to the case where $S=\mathrm{Spec}(A)$ is affine; if $K=k(\eta )$, the field of fractions of $A$, there exists by hypothesis and by virtue of (9.1.4) a finite extension $K^{\prime }$ of $K$ and a $K^{\prime }$-morphism $g^{\prime }:(X_{\eta }{)}_{(K^{\prime })}\to (Y_{\eta }{)}_{(K^{\prime })}$ satisfying $Q(g^{\prime })$, and $K^{\prime }$ is evidently the field of fractions of an integral $A$-algebra $A^{\prime }$ finite over $A$. If one sets $S^{\prime }=\mathrm{Spec}(A^{\prime })$, it then follows from (8.10.5) that there is a neighbourhood $U^{\prime }$ of the generic point ${\eta }^{\prime }$ of $S^{\prime }$ such that, if one sets $X^{\prime }=X{\otimes }_A{A}^{\prime }$, $Y^{\prime }=Y{\otimes }_A{A}^{\prime }$, there exists, for every $s^{\prime }\in U^{\prime }$, a morphism $X_{s^{\prime }}^{\prime }\to Y_{s^{\prime }}^{\prime }$ having the property $Q$. But the morphism $h:S^{\prime }\to S$ is finite, hence closed, and since $h^{-1}(\eta )={\eta }^{\prime }$, $h(U^{\prime })$ contains an open neighbourhood $U$ of $\eta$ in $S$; for every $s\in U$, there is therefore $s^{\prime }\in U^{\prime }$ such that $h(s^{\prime })=s$, and since $X_{s^{\prime }}^{\prime }=X_s{\otimes }_{k(s)}k(s^{\prime })$, $Y_{s^{\prime }}^{\prime }=Y_s{\otimes }_{k(s)}k(s^{\prime })$, the property $P(X_s,Y_s,k(s))$ is true for every $s\in U$.

Example (9.3.6).

Take for example for $Q$ the property of being an isomorphism. Then, by combining (9.3.5) and (9.3.3), one has the following property: the notations and hypotheses being those of (8.8.1), $S_{\alpha }$ being quasi-compact, $X_{\alpha }$ and $Y_{\alpha }$ of finite presentation over $S_{\alpha }$, in order that, for every $s\in S$, $X_s$ and $Y_s$ be geometrically isomorphic (9.1.4), it is necessary and sufficient that there exist $\lambda ⩾\alpha$ such that, for every $s_{\lambda }\in S_{\lambda }$, $X_{\lambda ,s_{\lambda }}$ and $Y_{\lambda ,s_{\lambda }}$ be geometrically isomorphic.

One has an analogous result when the preschemes one considers are equipped with "composition laws" of a certain kind $(0_{III},\mathrm{8.2.1})$, for example "preschemes in groups", "preschemes in rings", etc. $(0_{III},\mathrm{8.2.3})$. Then the preceding statement is still valid when by "isomorphism" one means isomorphisms of preschemes that are homomorphisms for the composition laws considered $(0_{III},\mathrm{8.2.2})$; it suffices here to use not only (8.10.5) but also (8.8.2, (i)), remarking that the notion of homomorphism for a composition law is expressed by writing that diagrams of morphisms of preschemes are commutative (it is of course necessary that the transition morphisms $X_{\mu }\to X_{\lambda }$ and $Y_{\mu }\to Y_{\lambda }$ for $\lambda ⩽\mu$ be homomorphisms for the composition laws envisaged).

One may also, instead of considering morphisms of preschemes as in (9.3.5), consider homomorphisms of Modules, using (9.1.6) in place of (9.1.5).

9.4. Constructibility of certain properties of modules

Notation (9.4.1).

In this number and the following ones up to the end of §9, we shall systematically use the following notation: given a morphism $f:X\to S$, we shall set, for every $s\in S$, $X_s=f^{-1}(s)=X{\times }_S\mathrm{Spec}(k(s))$; for every quasi-coherent ${\mathcal{O}}_X$-Module $\mathcal{F}$, ${\mathcal{F}}_s$ will denote the ${\mathcal{O}}_{X_s}$-Module $\mathcal{F}{\otimes }_{{\mathcal{O}}_S}k(s)$, and for every homomorphism $u:\mathcal{F}\to \mathcal{G}$ of $\mathcal{F}$ into a quasi-coherent ${\mathcal{O}}_X$-Module $\mathcal{G}$, $u_s:{\mathcal{F}}_s\to {\mathcal{G}}_s$ will be the morphism $p\ast (u)$, where $p$ is the canonical projection $X_s\to X$. For every section $g$ of $\mathcal{F}$ above $X$, one shall denote by $g_s$ the image of $g$ under the canonical homomorphism $\Gamma (X,\mathcal{F})\to \Gamma (X_s,{\mathcal{F}}_s)$. For every part $Z$ of $X$, one will denote by $Z_s$ the inverse image $p^{-1}(Z)=Z\cap X_s$ (I, 3.6.1). Finally, if $Y$ is a second $S$-prescheme and $h:X\to Y$ an $S$-morphism, one will denote by $h_s$ the morphism $h\times 1:X_s\to Y_s$.

Proposition (9.4.2).

Let $S$ be an integral prescheme of generic point $\eta$, $f:X\to S$ a morphism of finite presentation, $\mathcal{F}$, $\mathcal{G}$, $\mathcal{H}$ three quasi-coherent ${\mathcal{O}}_X$-Modules of finite presentation. Let $u:\mathcal{F}\to \mathcal{G}$, $v:\mathcal{G}\to \mathcal{H}$ be two homomorphisms of ${\mathcal{O}}_X$-Modules, and suppose that the sequence ${\mathcal{F}}_{\eta }\stackrel{u_{\eta }}{\to }{\mathcal{G}}_{\eta }\stackrel{v_{\eta }}{\to }{\mathcal{H}}_{\eta }$ is exact. Then there exists an open neighbourhood $U$ of $\eta$ in $S$ such that, for every $s\in U$, the sequence ${\mathcal{F}}_s\stackrel{u_s}{\to }{\mathcal{G}}_s\stackrel{v_s}{\to }{\mathcal{H}}_s$ is exact.

With the general notation of (9.2.1), this concerns the relation $P(X,\mathcal{F},\mathcal{G},\mathcal{H},u,v,k)$: "$X$ is an algebraic prescheme over the field $k$, $\mathcal{F}\to \mathcal{G}\to \mathcal{H}$ an exact sequence of quasi-coherent ${\mathcal{O}}_X$-Modules". Since, for every extension $k^{\prime }$ of $k$, the canonical projection $X_{(k^{\prime })}\to X$ is a faithfully flat morphism (2.2.13), condition (9.2.1, (i)) is satisfied (2.2.7). By virtue of (9.2.3), one may restrict to the case where $S$ is integral and Noetherian, in which case $X$ is Noetherian, and $\mathcal{F}$, $\mathcal{G}$, $\mathcal{H}$ are coherent ${\mathcal{O}}_X$-Modules. The hypothesis implies that there exists an open neighbourhood $U$ of $\eta$ in $S$ such that the sequence $\mathcal{F}|f^{-1}(U)\to \mathcal{G}|f^{-1}(U)\to \mathcal{H}|f^{-1}(U)$ is exact, by virtue of (8.5.8, (i)) applied following the general method of (8.1.2, a)), and one may therefore already suppose that the sequence $\mathcal{F}\to \mathcal{G}\to \mathcal{H}$ is exact; it evidently suffices to prove that one has $Ker(v_s)=(Kerv{)}_s$ and $Im(u_s)=(Imu{)}_s$ for every $s$ close to $\eta$ in $S$; consequently (taking account of the fact that the ${\mathcal{O}}_X$-Modules $\mathcal{F}$, $\mathcal{G}$, $\mathcal{H}$ are coherent and of $(0_I,\mathrm{5.3.4})$) one is reduced to proving the proposition in the particular case where the sequence $0\to \mathcal{F}\to \mathcal{G}\to \mathcal{H}\to 0$ is exact. But then there is an open set $U$ in $S$ containing $\eta$ and such that $\mathcal{H}|f^{-1}(U)$ is flat over $U$ (6.9.1); it then follows from (2.1.8) that for every $s\in U$, the sequence $0\to {\mathcal{F}}_s\to {\mathcal{G}}_s\to {\mathcal{H}}_s\to 0$ is exact, which completes the proof.

Corollary (9.4.3).

Let $S$ be an integral prescheme, of generic point $\eta$, $f:X\to S$ a morphism of finite presentation. Let $\mathcal{L}\bullet =({\mathcal{L}}^i{)}_{i\in \mathbb{Z}}$ be a complex of quasi-coherent ${\mathcal{O}}_X$-Modules of finite presentation. For every $i$, there exists an open neighbourhood $U$ of $\eta$ in $S$ such that the canonical homomorphisms

$$(\mathrm{9.4.3.1})({\mathcal{H}}^i(\mathcal{L}\bullet ){)}_s\to {\mathcal{H}}^i(\mathcal{L}{\bullet }_s)$$

are bijective for every $s\in U$.

One may evidently restrict to a complex with three terms of degrees $-1,0,+1$: $\mathcal{M}\stackrel{u}{\to }\mathcal{N}\stackrel{v}{\to }\mathcal{P}$ with $v\circ u=0$; and to $i=0$; the homomorphism to consider is then the canonical homomorphism $(Kerv/Imu{)}_s\to Ker(v_s)/Im(u_s)$. Using (8.9.1) and (8.5.2, (i)), one sees that one may reduce to the case where $S$ (hence also $X$) is Noetherian, and consequently $\mathcal{M}$, $\mathcal{N}$, $\mathcal{P}$ are coherent ${\mathcal{O}}_X$-Modules; then $Im(u)$ and $Ker(v)$ are also coherent $(0_I,\mathrm{5.3.4})$ and moreover there exists a neighbourhood $U$ of $\eta$ such that for $s\in U$, one has $Ker(v_s)=(Ker(v){)}_s$ and $Im(u_s)=(Im(u){)}_s$ (9.4.2); the conclusion then results from (9.4.2) applied to the exact sequence $0\to Im(u)\to Ker(v)\to Ker(v)/Im(u)\to 0$, taking account of the fact that ${\mathcal{O}}_{\eta }=k(\eta )$ (since $S$ is integral) and consequently the sequence

  0 → (Im u)_η → (Ker v)_η → (Ker v / Im u)_η → 0

is exact.

Proposition (9.4.4).

Let $f:X\to S$ be a morphism of finite presentation, $\mathcal{F}$, $\mathcal{G}$, $\mathcal{H}$ three quasi-coherent ${\mathcal{O}}_X$-Modules of finite presentation. Let $u:\mathcal{F}\to \mathcal{G}$, $v:\mathcal{G}\to \mathcal{H}$ be two homomorphisms of ${\mathcal{O}}_X$-Modules. Then the set $E$ of $s\in S$ such that the sequence ${\mathcal{F}}_s\stackrel{u_s}{\to }{\mathcal{G}}_s\stackrel{v_s}{\to }{\mathcal{H}}_s$ is exact is locally constructible.

Taking account of (9.2.3), one must establish that the property $P$ considered in (9.4.2) is constructible. One has already remarked in the proof of (9.4.2) that condition (9.2.1, (i)) is satisfied for this property, and it remains to verify condition (9.2.1, (ii)). Suppose then that $S$ is integral Noetherian, of generic point $\eta$, and let us prove that $E$ or $S-E$ is a neighbourhood of $\eta$. If $\eta \in E$, our assertion follows from (9.4.2), and one may therefore restrict to the case where $\eta \notin E$, that is, the sequence ${\mathcal{F}}_{\eta }\stackrel{u_{\eta }}{\to }{\mathcal{G}}_{\eta }\stackrel{v_{\eta }}{\to }{\mathcal{H}}_{\eta }$ is not exact. Let us distinguish two cases.

1° Set $w=v\circ u$, and suppose first that $w_{\eta }=v_{\eta }\circ u_{\eta }\ne 0$. Since $\mathcal{F}$, $\mathcal{G}$, $\mathcal{H}$ are coherent, the same is true of $\mathcal{N}=Ker(w)$ $(0_I,\mathrm{5.3.4})$; it then follows from (9.4.2) applied to the exact sequence $0\to \mathcal{N}\to \mathcal{F}\to \mathcal{F}$ that there is a neighbourhood $U$ of $\eta$ in $S$ such that, for $s\in S$, $Ker(w_s)={\mathcal{N}}_s$; by restricting $S$, one may therefore suppose this relation verified for every $s\in S$. Let $j$ be the canonical injection $\mathcal{N}\to \mathcal{F}$, and set $\mathcal{M}=Coker(j)$; the right-exactness of the functor $\mathcal{F}\mapsto {\mathcal{F}}_s$ entails that ${\mathcal{M}}_s=Coker(j_s)$ for every $s\in S$. The hypothesis $w_{\eta }\ne 0$ means that ${\mathcal{M}}_{\eta }\ne 0$; since $\mathcal{M}$ is coherent $(0_I,\mathrm{5.3.4})$, it follows from (1.8.6) that there is an open neighbourhood $U$ of $\eta$ in $S$ such that ${\mathcal{M}}_s\ne 0$ for $s\in U$, hence $w_s\ne 0$ for $s\in U$, and a fortiori $S-E$ is a neighbourhood of $\eta$.

2° Suppose that $w_{\eta }=0$; by virtue of (8.5.2, (i)), applied following the general method of (8.1.2, a)), there exists an open neighbourhood $U$ of $\eta$ such that $w|f^{-1}(U)=0$; replacing $S$ by $U$, one may already suppose $w=0$ in $X$. Then $\mathcal{F}\stackrel{u}{\to }\mathcal{G}\stackrel{v}{\to }\mathcal{H}$ is a complex with three terms $\mathcal{L}\bullet$, to which one may apply (9.4.3); by hypothesis one has $({\mathcal{H}}^0(\mathcal{L}\bullet ){)}_{\eta }={\mathcal{H}}^0(\mathcal{L}{\bullet }_{\eta })\ne 0$, and ${\mathcal{H}}^0(\mathcal{L}\bullet )$ is coherent $(0_I,\mathrm{5.3.4}and\mathrm{5.3.3})$, hence it follows from (1.8.6) that there is an open neighbourhood $U$ of $\eta$ such that $({\mathcal{H}}^0(\mathcal{L}\bullet ){)}_s\ne 0$ for every $s\in U$; but as one may suppose that ${\mathcal{H}}^0(\mathcal{L}\bullet {)}_s=({\mathcal{H}}^0(\mathcal{L}\bullet ){)}_s$ for $s\in U$ by (9.4.3), one sees again that $S-E$ is a neighbourhood of $\eta$ in $S$.

Corollary (9.4.5).

Let $f:X\to S$ be a morphism of finite presentation, $\mathcal{F}$, $\mathcal{G}$ two quasi-coherent ${\mathcal{O}}_X$-Modules of finite presentation, $u:\mathcal{F}\to \mathcal{G}$ a homomorphism of ${\mathcal{O}}_X$-Modules. Then the set of $s\in S$ such that $u_s$ is injective (resp. surjective, bijective, zero) is locally constructible.

It suffices to apply (9.4.4) to the sequences $0\to \mathcal{F}\stackrel{u}{\to }\mathcal{G}$, $\mathcal{F}\stackrel{u}{\to }\mathcal{G}\to 0$, $0\to \mathcal{F}\stackrel{u}{\to }\mathcal{G}\to 0$, $\mathcal{F}\stackrel{u}{\to }\mathcal{G}\stackrel{1_{\mathcal{G}}}{\to }\mathcal{G}$.

Corollary (9.4.6).

Let $f:X\to S$ be a morphism of finite presentation, $\mathcal{F}$ a quasi-coherent ${\mathcal{O}}_X$-Module of finite presentation. Let $h$ be a section of $\mathcal{F}$ above $X$; for every $s\in S$, let $h_s$ be the corresponding section of ${\mathcal{F}}_s$ above $X_s$ (for the projection morphism $X_s\to X$). Then the set of $s\in S$ such that $h_s=0$ is locally constructible.

It suffices to note that $h$ corresponds to a homomorphism $u:{\mathcal{O}}_X\to \mathcal{F}$ $(0_I,\mathrm{5.1.1})$ and $h_s$ to the homomorphism $u_s$.

Proposition (9.4.7).

Let $f:X\to S$ be a morphism of finite presentation, $\mathcal{F}$ a quasi-coherent ${\mathcal{O}}_X$-Module of finite presentation. The set $E$ (resp. $E^{\prime }$) of $s\in S$ such that ${\mathcal{F}}_s$ is a locally free ${\mathcal{O}}_{X_s}$-Module (resp. locally free of rank $n$) is locally constructible.

If $X$ is an algebraic prescheme over a field $k$, $\mathcal{F}$ a coherent ${\mathcal{O}}_X$-Module, $k^{\prime }$ an extension of $k$, then, for $\mathcal{F}$ to be locally free (resp. locally free of rank $n$), it is necessary and sufficient that the same be true of $\mathcal{F}{\otimes }_k{k}^{\prime }$, since the projection $X_{(k^{\prime })}\to X$ is a faithfully flat morphism (2.2.7). In other words, condition (9.2.1, (i)) is verified for the properties whose constructibility one wishes to prove, and it remains to verify (9.2.1, (ii)); one may therefore again suppose that $S$ is affine, Noetherian, and integral. There are once more four cases to envisage.

1° $\eta \in E$. It follows from (8.5.5), applied following the general method of (8.1.2, a)), that there exists an open neighbourhood $U$ of $\eta$ in $S$ such that $\mathcal{F}|f^{-1}(U)$ is locally free; a fortiori ${\mathcal{F}}_s$ is locally free for every $s\in U$.

2° $\eta \in E^{\prime }$. Same reasoning as in 1°.

3° $\eta \in S-E$. Since ${\mathcal{F}}_{\eta }$ is a coherent ${\mathcal{O}}_{X_{\eta }}$-Module, to say that it is not locally free is equivalent to saying that it is not flat over ${\mathcal{O}}_{X_{\eta }}$ (Bourbaki, Alg. comm., chap. II, §5, n° 2, cor. 2 of th. 1). The fact that $S-E$ is a neighbourhood of $\eta$ will therefore result from the more general lemma below (applied to the case where $g$ is the identity).

Lemma (9.4.7.1).

Let $S$ be an integral Noetherian prescheme, of generic point $\eta$, $X$, $Y$ two $S$-preschemes of finite type over $S$, $g:X\to Y$ an $S$-morphism, $\mathcal{F}$ a coherent ${\mathcal{O}}_X$-Module. If ${\mathcal{F}}_{\eta }$ is not $g_{\eta }$-flat, then there exists an open neighbourhood $U$ of $\eta$ in $S$ such that for every $s\in U$, ${\mathcal{F}}_s$ is not $g_s$-flat.

Taking account of (2.1.2) and of Bourbaki, Alg. comm., chap. I, §2, n° 3, Remark 1, the hypothesis means that there exists a non-empty open set $V$ of $Y_{\eta }$ and an injective homomorphism $v:\mathcal{M}\to \mathcal{N}$ of coherent ${\mathcal{O}}_V$-Modules, such that the homomorphism $1\otimes v:{\mathcal{F}}_{\eta }{\otimes }_{{\mathcal{O}}_V}\mathcal{M}\to {\mathcal{F}}_{\eta }{\otimes }_{{\mathcal{O}}_V}\mathcal{N}$ is not injective. One has $V=Y_{\eta }\cap W$, where $W$ is open in $Y$ (I, 3.6.1), and it follows from (8.5.2, (i) and (ii)), applied following the method of (8.1.2, a)), that there exists an open neighbourhood U_0 of $\eta$ in $S$, two coherent ${\mathcal{O}}_Z$-Modules

${\mathcal{M}}^{\prime }$, ${\mathcal{N}}^{\prime }$ (where $Z=W\cap h^{-1}(U_0)$, $h:Y\to S$ being the structure morphism) and an ${\mathcal{O}}_Z$-homomorphism $u:{\mathcal{M}}^{\prime }\to {\mathcal{N}}^{\prime }$ such that ${\mathcal{M}}_{\eta }^{\prime }=\mathcal{M}$, ${\mathcal{N}}_{\eta }^{\prime }=\mathcal{N}$ and $v=u_{\eta }$; one may therefore suppose U_0 taken such that for $s\in U_0$, $u_s:{\mathcal{M}}_s^{\prime }\to {\mathcal{N}}_s^{\prime }$ is injective (9.4.5). But for every $s\in U_0$, the homomorphism $1\otimes u_s:{\mathcal{F}}_s{\otimes }_{{\mathcal{O}}_{Y_s}}{\mathcal{M}}_s^{\prime }\to {\mathcal{F}}_s{\otimes }_{{\mathcal{O}}_{Y_s}}{\mathcal{N}}_s^{\prime }$ is none other than $(1\otimes u{)}_s$; the hypothesis that $(1\otimes u{)}_{\eta }$ is non-injective therefore entails (9.4.5) the existence of a non-empty open set $U\subset U_0$ such that for every $s\in U$, $(1\otimes u{)}_s$ is non-injective, and consequently ${\mathcal{F}}_s$ is not $g_s$-flat for every $s\in U$.

4° $\eta \in S-E^{\prime }$. It is clear that $S-E\subset S-E^{\prime }$, and if $\eta \in S-E$, $S-E^{\prime }$ is a fortiori a neighbourhood of $\eta$ by 3°. Suppose therefore that $\eta \in E$, hence ${\mathcal{F}}_{\eta }$ locally free; these hypotheses entail that $X_{\eta }$ is disconnected, and that the ranks of the locally free ${\mathcal{O}}_{X_{\eta }}$-Module ${\mathcal{F}}_{\eta }$ are not the same on the various connected components of $X_{\eta }$. Now it follows from (8.4.2), applied following the method of (8.1.2, a)), that one may suppose (replacing $S$ by an open neighbourhood of $\eta$) that $X$ and $X_{\eta }$ have the same number of connected components, the connected components of $X_{\eta }$ being the intersections of $X_{\eta }$ with the connected components of $X$. The conclusion then results from the reasoning made in 2°, applied to each of the connected components of $X$ (which are finite in number).

Remark (9.4.7.2).

The lemma (9.4.7.1) will be generalized later and freed of Noetherian hypotheses (11.2.8).

Proposition (9.4.8).

Let $S$ be a locally Noetherian prescheme, $f:X\to S$ a morphism of finite type, $\mathcal{F}$ a coherent ${\mathcal{O}}_X$-Module. Suppose that for every $s\in S$, $X_s$ is a locally integral prescheme. Then the set $E$ (resp. $E^{\prime }$) of $s\in S$ such that ${\mathcal{F}}_s$ is a torsion ${\mathcal{O}}_{X_s}$-Module (resp. a torsion-free ${\mathcal{O}}_{X_s}$-Module) is locally constructible.

One may evidently suppose $S=\mathrm{Spec}(A)$ affine and Noetherian and prove that $E$ (resp. $E^{\prime }$) is then constructible by using the criterion $(0_{III},\mathrm{9.2.3})$; replacing $S$ by the reduced closed sub-prescheme of $S$ having an irreducible closed part $Y$ of $S$ as underlying space, and noting (I, 3.6.4) that for $s\in Y$, the fibre $(X_{(Y)}{)}_s$ identifies canonically with $X_s$ and the sheaf $(\mathcal{F}{\otimes }_{{\mathcal{O}}_S}{\mathcal{O}}_Y{)}_s$ with ${\mathcal{F}}_s$, one sees that one is reduced to the case where $S$ is integral of generic point $\eta$, and to proving that $E$ or $S-E$ (resp. $E^{\prime }$ or $S-E^{\prime }$) is a neighbourhood of $\eta$ in $S$. Note moreover that $X$ is a finite union of affine open sets $U_i$ of finite type over $S$, and each of $(U_i{)}_s$ is induced on an open set of $X_s$, hence locally integral; in addition, if $(U_i{)}_{\eta }$ is empty (resp. non-empty), one knows that $(U_i{)}_s$ is also empty (resp. non-empty) in a neighbourhood of $\eta$ (9.2.6). One may therefore suppose all the $(U_i{)}_{\eta }$ non-empty and integral, and to say that ${\mathcal{F}}_s$ is torsion (resp. torsion-free) is equivalent to saying that each of the ${\mathcal{F}}_s|(U_i{)}_s$ is torsion (resp. torsion-free). One is therefore reduced to the case where $X=\mathrm{Spec}(B)$ is affine, and $\mathcal{F}=\tilde{M}$, where $M$ is a $B$-module of finite type; one sets $B_s=B{\otimes }_{A}k(s)$, $M_s=M{\otimes }_{A}k(s)$, and one may suppose $B_{\eta }$ integral. We have four cases to envisage.

1° $\eta \in E$; $M_{\eta }$ is then a torsion $B_{\eta }$-module of finite type, and there is consequently $h\ne 0$ in $B_{\eta }$ such that $h{M}_{\eta }=0$; by virtue of (8.5.2, (i)), applied following the method

of (8.1.2, a)), one may (replacing $S$ if necessary by a neighbourhood of $\eta$) suppose that $h\in \Gamma (X_{\eta },{\mathcal{O}}_{X_{\eta }})$ is of the form $g_{\eta }$, where $g\in \Gamma (X,{\mathcal{O}}_X)$; let $u:\mathcal{F}\to \mathcal{F}$ be the endomorphism of $\mathcal{F}$ defined by multiplication by $g$; by hypothesis, one has $u_{\eta }=0$, hence (9.4.5) the endomorphism $u_s:{\mathcal{F}}_s\to {\mathcal{F}}_s$, defined by multiplication by $g_s$, is zero in a neighbourhood of $\eta$. On the other hand, let $v:{\mathcal{O}}_X\to {\mathcal{O}}_X$ be the endomorphism defined by multiplication by $g$; since $B_{\eta }$ is integral and $h=g_{\eta }\ne 0$, $v_{\eta }$ is injective, and it therefore follows from (9.4.5) that $v_s$ is an injective endomorphism of ${\mathcal{O}}_{X_s}$ for $s$ close to $\eta$, in other words, $g_s$ is an ${\mathcal{O}}_{X_s}$-regular element for these values of $s$; hence ${\mathcal{F}}_s$ is torsion in a neighbourhood of $\eta$.

2° $\eta \in S-E$. To say that a $B_{\eta }$-module $M_{\eta }$ of finite type is not a torsion module means that its quotient $M_{\eta }/T$ by its torsion sub-module is $\ne 0$, and since it is a torsion-free $B_{\eta }$-module of finite type, it is isomorphic to a sub-module of a $B_{\eta }$-module $B_{\eta }^n$; there is consequently a homomorphism $w:M_{\eta }\to B_{\eta }^n$ which is $\ne 0$. Applying (8.5.2, (i)) following the method of (8.1.2, a)), one deduces (replacing $S$ if necessary by a neighbourhood of $\eta$) that there exists a homomorphism $v:\mathcal{F}\to {\mathcal{O}}_X^n$ such that $v_{\eta }=w$. The hypothesis $v_{\eta }\ne 0$ therefore entails (9.4.5) that $v_s\ne 0$ in a neighbourhood of $\eta$, and since $X_s$ is locally integral, ${\mathcal{F}}_s$ is not torsion for these values of $s$.

3° $\eta \in E^{\prime }$. Since $M_{\eta }$ is a torsion-free $B_{\eta }$-module of finite type, there exists an injective homomorphism $w:M_{\eta }\to B_{\eta }^n$. Using (8.5.2, (i)) and (9.4.5) as in 2° (restricting $S$ if necessary), one deduces that there exists a homomorphism $v:\mathcal{F}\to {\mathcal{O}}_X^n$ such that $v_{\eta }=w$ and that for $s$ close to $\eta$, $v_s:{\mathcal{F}}_s\to {\mathcal{O}}_{X_s}^n$ is injective; for these values of $s$, ${\mathcal{F}}_s$ is therefore torsion-free.

4° $\eta \in S-E^{\prime }$. Let $T$ be the torsion sub-module of $M_{\eta }$; by hypothesis $T\ne 0$, and $T$ is of finite type since $M_{\eta }$ is Noetherian. Using this time (8.5.2, (i) and (ii)) one sees (restricting $S$ if necessary) that there exists a coherent ${\mathcal{O}}_X$-Module $\mathcal{G}$ and an injective homomorphism $u:\mathcal{G}\to \mathcal{F}$ such that ${\mathcal{G}}_{\eta }=T$ and $u_{\eta }$ is the canonical injection $T\to {\mathcal{F}}_{\eta }$. It then follows from 1° and from (1.8.6) that in a neighbourhood of $\eta$, ${\mathcal{G}}_s$ is a torsion ${\mathcal{O}}_{X_s}$-Module $\ne 0$, and on the other hand it follows from (9.4.5) that in a neighbourhood of $\eta$, $u_s$ is injective. One concludes that in a neighbourhood of $\eta$, the torsion sub-Module of ${\mathcal{F}}_s$ is non-zero. C.Q.F.D.

Remark (9.4.9).

The property "$X$ is a locally integral algebraic $k$-prescheme" does not verify condition (9.2.1, (i)), and it is therefore not certain that the statement (9.4.8) remains valid when one makes no hypothesis on $S$ and one supposes only that $f$ is a morphism of finite presentation and $\mathcal{F}$ an ${\mathcal{O}}_X$-Module of finite presentation. Let us nevertheless consider the following particular case: S_0 being a locally Noetherian prescheme, let $f_0:X_0\to S_0$ be a morphism of finite type, such that the fibres $(X_0{)}_{s_0}$ are locally integral (for every $s_0\in S_0$), and ${\mathcal{F}}_0$ a coherent ${\mathcal{O}}_{X_0}$-Module; let $g:S\to S_0$ be an arbitrary morphism, set $X=X_0{\times }_{S_0}S$, $\mathcal{F}={\mathcal{F}}_0{\otimes }_{{\mathcal{O}}_{S_0}}{\mathcal{O}}_S$, $f=f_0\times 1_S:X\to S$, and suppose that for every $s\in S$, the fibre $X_s$ is still locally integral. Then the set $E$ (resp. $E^{\prime }$) of $s\in S$ such that ${\mathcal{F}}_s$ is torsion (resp. torsion-free)

is still locally constructible. Indeed, let $s\in S$ and let $s_0=g(s)$; it will suffice (taking into account (1.8.2)) to prove that, for ${\mathcal{F}}_s$ to be torsion (resp. torsion-free), it is necessary and sufficient that $({\mathcal{F}}_0{)}_{s_0}$ be so. Now, $Supp({\mathcal{F}}_s)$ is the inverse image of $Supp(({\mathcal{F}}_0{)}_{s_0})$ by the projection $p:X_s\to (X_0{)}_{s_0}$ (I, 9.1.13); since $p$ is faithfully flat and quasi-compact, to say that $Supp({\mathcal{F}}_s)$ contains a maximal point of $X_s$ is equivalent to saying that $Supp(({\mathcal{F}}_0{)}_{s_0})$ contains a maximal point of $(X_0{)}_{s_0}$ (1.1.5 and 2.3.4); whence our assertion concerning the set $E$ (I, 7.4.6). If the torsion sub-Module $\mathcal{T}$ of $({\mathcal{F}}_0{)}_{s_0}$ is non-zero, $\mathcal{T}{\otimes }_{k(s_0)}k(s)$ (which is torsion by what precedes) is non-zero and identifies with a sub-Module of ${\mathcal{F}}_s$ (2.2.7), hence the torsion sub-Module of ${\mathcal{F}}_s$ is non-zero. Finally, if $({\mathcal{F}}_0{)}_{s_0}$ is torsion-free, one may suppose (by considering an affine open set of $(X_0{)}_{s_0}$) that $({\mathcal{F}}_0{)}_{s_0}$ is isomorphic to a sub-Module of a ${\mathcal{O}}_{(X_0{)}_{s_0}}^n$, hence ${\mathcal{F}}_s$ is isomorphic to a sub-Module of an ${\mathcal{O}}_{X_s}^n$ (2.2.7), and this establishes our assertion concerning $E^{\prime }$.

9.5. Constructibility of topological properties

Proposition (9.5.1).

Let $f:X\to S$ be a morphism of finite presentation, $Z$ a locally constructible part of $X$. Then the set $E$ of $s\in S$ such that $Z_s\ne \varnothing$ is locally constructible.

Indeed, one has $E=f(Z)$, and it suffices to apply Chevalley's theorem (1.8.4).

Corollary (9.5.2).

If $Z^{\prime }$, Z'' are two locally constructible parts of $X$, the set of $s\in S$ such that $Z_s^{\prime }\subset Z_s^{\prime \prime }$ (resp. $Z_s^{\prime }=Z_s^{\prime \prime }$) is locally constructible.

Indeed, the relation $Z_s^{\prime }\subset Z_s^{\prime \prime }$ is equivalent to $(Z^{\prime }\cap (\complement Z^{\prime \prime }){)}_s=\varnothing$, and $Z^{\prime }\cap \complement Z^{\prime \prime }$ is locally constructible.

Proposition (9.5.3).

Let $f:X\to S$ be a morphism of finite presentation, $Z$, $Z^{\prime }$ two locally constructible parts of $X$ such that $Z\subset Z^{\prime }$. Then the set $E$ of $s\in S$ such that $Z_s$ is dense in $Z_s^{\prime }$ is locally constructible in $S$.

One must verify the two conditions of (9.2.2, (ii)). As for the first, consider an algebraic prescheme $X$ over a field $k$, two constructible parts $Z$, $Z^{\prime }$ of $X$ such that $Z\subset Z^{\prime }$, and an extension $k^{\prime }$ of $k$. Then the canonical projection $p:X_{(k^{\prime })}\to X$ is a faithfully flat and quasi-compact morphism, and one therefore has $p^{-1}(\overline{Z})=\overline{p^{-1}(Z)}$ and $p^{-1}({\overline{Z}}^{\prime })=\overline{p^{-1}(Z^{\prime })}$ by virtue of (2.3.10); since $p$ is surjective, the relation $\overline{Z}={\overline{Z}}^{\prime }$ is equivalent to $\overline{p^{-1}(Z)}=\overline{p^{-1}(Z^{\prime })}$.

Let us now verify the second condition, and suppose therefore $S$ affine, Noetherian, and integral, of generic point $\eta$. Let us distinguish two cases.

1° $\eta \in S-E$, in other words, $Z_{\eta }$ is not dense in $Z_{\eta }^{\prime }$; there exists therefore in $X$ an open set $V$ such that $V\cap Z_{\eta }=\varnothing$ and $V\cap Z_{\eta }^{\prime }\ne \varnothing$. As $X$ is Noetherian, $V$ is locally constructible, hence so is $V\cap Z$, and by virtue of (9.5.1), there is a neighbourhood $U$ of $\eta$ in $S$ such that for every $s\in U$, one has $(V\cap Z{)}_s=\varnothing$ and $(V\cap Z^{\prime }{)}_s\ne \varnothing$; this entails that $Z_s$ is not dense in $Z_s^{\prime }$ for $s\in U$, in other words $U\subset S-E$.

2° $\eta \in E$, hence $Z_{\eta }$ is dense in $Z_{\eta }^{\prime }$. Let us first show that one may suppose $Z^{\prime }$ closed. Indeed, $Z_{\eta }$ is dense in $\overline{Z_{\eta }^{\prime }}$ (closure taken in $X_{\eta }$); set $V_{\eta }=X_{\eta }-\overline{Z_{\eta }^{\prime }}$, which is open in $X_{\eta }$ and does not meet $Z_{\eta }$; one may suppose $V_{\eta }$ of the form $V\cap X_{\eta }$, where $V$ is open (hence constructible) in $X$, and the hypothesis $V_{\eta }\cap Z_{\eta }^{\prime }=\varnothing$ then entails $V_s\cap Z_s^{\prime }=\varnothing$ for every $s$ close to $\eta$ by virtue of (9.5.1). Replacing $S$ by an open neighbourhood of $\eta$, one may therefore suppose that $V\cap Z^{\prime }=\varnothing$, hence $V\cap {\overline{Z}}^{\prime }=\varnothing$ (closure taken in $X$), and consequently $({\overline{Z}}^{\prime }{)}_{\eta }=\overline{Z_{\eta }^{\prime }}$, whence our assertion. The set ${\overline{Z}}^{\prime }$ is then the union of its irreducible components in finite number, and by restricting $S$ again to a neighbourhood of $\eta$, one may suppose that all the irreducible components $Z_i^{\prime }$ of ${\overline{Z}}^{\prime }$ meet $X_{\eta }$, whence it follows that $X_{\eta }$ contains the generic points of the $Z_i^{\prime }$ $(0_I,\mathrm{2.1.8})$. To say that $Z_s$ is dense in $Z_s^{\prime }$ is then equivalent to saying that each of the $(Z\cap Z_i^{\prime }{)}_s$ is dense in $(Z_i^{\prime }{)}_s$, and one is thus reduced to the case where ${\overline{Z}}^{\prime }$ is irreducible. Replacing then if necessary $X$ by the reduced sub-prescheme having ${\overline{Z}}^{\prime }$ as underlying space, one sees that one may suppose that $Z^{\prime }=X$ and that $X$ is integral and dominates $S$. Finally, by covering $X$ by a finite number of affine open sets $W_j$ and replacing $Z$ by $Z\cap W_j$, one may suppose that $X=\mathrm{Spec}(A)$, where $A$ is an integral Noetherian ring. Since $X_{\eta }$ is integral Noetherian and $Z_{\eta }$ is constructible in $X_{\eta }$ and dense in $X_{\eta }$, $Z_{\eta }$ contains a non-empty open set of $X_{\eta }$ $(0_{III},\mathrm{9.2.2})$, which one may suppose of the form $(D(t){)}_{\eta }$, where $t$ is an element $\ne 0$ of $A$. Replacing $S$ if necessary by a neighbourhood of $\eta$, one may moreover, by virtue of the relation $(D(t){)}_{\eta }\subset Z_{\eta }$, suppose that $D(t)\subset Z$ (9.5.2). Finally, since the homothety of ratio $t$ in ${\mathcal{O}}_X$ is injective, it follows from (9.4.5) that for $s$ close to $\eta$, $t_s$ is ${\mathcal{O}}_{X_s}$-regular, hence $(X_s{)}_{t_s}$ is dense in $X_s$, and a fortiori the same is true of $Z_s$, which contains $(X_s{)}_{t_s}$.