Éléments de Géométrie Algébrique — English

§12. Study of the fibres of flat morphisms of finite presentation

12.0. Introduction

Throughout this section we shall use the general notations of (9.4.1).

(12.0.1)

Given a morphism $f:X\to Y$ locally of finite presentation, we saw (9.9) that for certain local properties $P$ of preschemes over fields or of Modules on such preschemes, the set $E$ of $x\in X$ such that the property $P$ holds for the fibre $X_{f(x)}$ at the point $x$ of that fibre is locally constructible in $X$. We propose to show that, for most of these properties, if one supposes moreover that the morphism $f$ is flat, then the set $E$ is even open in $X$. Likewise ((9.2) through (9.8)) we have shown that if $f$ is of finite presentation, and if this time $P$ denotes certain global properties of preschemes over fields or of Modules on such preschemes, then the set $F$ of $y\in Y$ such that the property $P$ holds for the fibre $X_y$ is locally constructible in $Y$. We shall show that if one supposes moreover that the morphism $f$ is proper and flat, then $F$ is even open in $Y$.

(12.0.2)

The general method of proof of the properties in question comprises three steps. One first reduces to the case where $Y$ is affine and $X$ of finite presentation; then:

A) Using (8.9.1) (and possibly other results of §8) and (11.2.6), one reduces to the case where $X$ and $Y$ are Noetherian.

B) One applies the results of §9 recalled in (12.0.1) proving that $E$ (resp. $F$) is constructible.

C) To see that $E$ is open, it suffices, by virtue of $(0_{III},\mathrm{9.2.5})$, to show that if $x\in E$, then every generization $x^{\prime }$ of $x$ also belongs to $E$. Using (II, 7.1.7), one sees, since $Y$ is Noetherian, that there exists a spectrum of a discrete valuation ring Y_1 and a morphism $h:Y_1\to X$ such that, if $y_1$ (resp. $y_1^{\prime }$) is the closed point (resp. the generic point) of Y_1, one has $h(y_1)=x$, $h(y_1^{\prime })=x^{\prime }$. One then makes the base change $g=f\circ h:Y_1\to Y$; taking into account the results of §§4 and 6 on locally Noetherian preschemes over fields and changes of base field, one is reduced to proving the assertion in question for a point $x_1$ of $X_1=X{\times }_Y{Y}_1$ above $y_1$ and for a generization $x_1^{\prime }$ of $x_1$ above $y_1^{\prime }$. Since $Y_1=\mathrm{Spec}(A)$, where $A$ is a discrete valuation ring with uniformizer $t$, one must in the end, for an $A$-module $M$, prove a property of $M$, knowing that $M/tM$ has the same property and that $t$ is $M$-regular (which follows from the flatness hypothesis); for this one uses the results of (3.4) and (5.12). One proceeds in the same way for the set $F\subset Y$.

12.1. Local properties of the fibres of a flat morphism locally of finite presentation

Theorem (12.1.1).

Let $f:X\to Y$ be a morphism locally of finite presentation, $\mathcal{F}$ an ${\mathcal{O}}_X$-Module that is $f$-flat and of finite presentation, $\Phi$ a finite subset of $\mathbb{Z}\cup \pm \infty$, $k$ an integer. The following subsets of $X$ are open:

(i) The set of $x\in X$ such that the dimensions of the associated prime cycles of ${\mathcal{F}}_{f(x)}$ containing $x$ are elements of $\Phi$.

(ii) The set of $x\in X$ such that the associated prime cycles of ${\mathcal{F}}_{f(x)}$ containing $x$ all have dimension $k$.

(iii) The set of $x\in X$ such that $x$ belongs to no embedded associated prime cycle of ${\mathcal{F}}_{f(x)}$.

(iv) The set of $x\in X$ such that ${\mathcal{F}}_{f(x)}$ is equidimensional at the point $x$ and possesses property $(S_k)$ at the point $x$ (5.7.2).

(v) The set of $x\in X$ such that $coprof(({\mathcal{F}}_{f(x)}{)}_x)\le k$ (0, 16.4.9).

(vi) The set of $x\in X$ such that ${\mathcal{F}}_{f(x)}$ is a Cohen-Macaulay ${\mathcal{O}}_X$-Module at the point $x$ (5.7.1).

(vii) The set of $x\in X$ such that ${\mathcal{F}}_{f(x)}$ is geometrically reduced at the point $x$ (4.6.22).

(viii) The set of $x\in X$ such that ${\mathcal{F}}_{f(x)}$ is geometrically pointwise integral at the point $x$ (4.6.22).

The questions being local on $X$, one reduces first to the case where $X=\mathrm{Spec}(B)$ and $Y=\mathrm{Spec}(A)$ are affine, with $f$ a morphism of finite presentation. There then exists a Noetherian sub-ring A_0 of $A$, an A_0-prescheme of finite type X_0, and a coherent ${\mathcal{O}}_{X_0}$-Module ${\mathcal{F}}_0$ such that ${\mathcal{F}}_0{\otimes }_{A_0}A$ is isomorphic to $\mathcal{F}$ (8.9.1); in addition, by virtue of (11.2.6), one may suppose that ${\mathcal{F}}_0$ is Y_0-flat (with $Y_0=\mathrm{Spec}(A_0)$). If $h:X\to X_0$ is the canonical projection, the set $E$ of points $x\in X$ where one of properties (i) to (viii) holds is equal to $h^{-1}(E_0)$, where E_0 is the set of $x_0\in X_0$ where the corresponding property for ${\mathcal{F}}_0$ holds: this follows, for properties (i) to (iii), from (4.2.7); for properties (iv) to (vi), from (6.7.1); for properties (vii) and (viii), from (4.7.11).

One is thus reduced to the case where $A$ and $B$ are Noetherian, $f$ of finite type, and $\mathcal{F}=\tilde{M}$, where $M$ is a $B$-module of finite type. By virtue of (9.9.2), one knows that the set $E$ is constructible for all the properties considered, and there remains in each case step C) of (12.0.2), where one must prove that $E$ is stable under generization.

(12.1.1.1)

Let us begin with case (iii), which is the simplest. Let $x^{\prime }\ne x$ be a generization of $x$ in $X$. Set $y=f(x)$, $y^{\prime }=f(x^{\prime })$; there exists a spectrum of a discrete valuation ring Y_1 and a morphism $u:Y_1\to X$ such that, if $y_1$ and $y_1^{\prime }$ are the closed point and the generic point of Y_1, one has $u(y_1)=x$ and $u(y_1^{\prime })=x^{\prime }$ (II, 7.1.7); set $g=f\circ u:Y_1\to Y$; if $X_1=X{\times }_Y{Y}_1$, there exists a Y_1-section $u_1:Y_1\to X_1$ such that $u=g_1\circ u_1$, where $g_1:X_1\to X$ is the canonical projection. Setting $x_1=u_1(y_1)$, $x_1^{\prime }=u_1(y_1^{\prime })$, one therefore has $g_1(x_1)=x$ and $g_1(x_1^{\prime })=x^{\prime }$, and $x_1^{\prime }$ is a generization of $x_1$. Applying (4.2.7) again, one sees

that one is reduced to the case where $Y=\mathrm{Spec}(A)$ is the spectrum of a discrete valuation ring, $y$ being the closed point and $y^{\prime }$ the generic point of $Y$. If $t$ is a uniformizer of $A$, the hypothesis that $\mathcal{F}$ is $f$-flat entails that $t$ is $\mathcal{F}$-regular $(0_I,\mathrm{6.3.4})$. By hypothesis none of the embedded associated prime cycles of $\mathcal{F}/t\mathcal{F}$ contains $x$. Then, it follows from (3.4.4) that $x$ belongs to no embedded associated prime cycle of $\mathcal{F}$, and the same is therefore true of every generization of $x$. In particular, $x^{\prime }$ belongs to no embedded associated prime cycle of $\mathcal{F}$, nor a fortiori to any of those associated to ${\mathcal{F}}_{y^{\prime }}$ ($X_{y^{\prime }}$ being a sub-prescheme induced on an open set of $X$).

(12.1.1.2)

Let us consider next cases (iv) and (v) ((vi) is only a special case of (v)). One proceeds as above (using (6.7.1) instead of (4.2.7)) and one is reduced to the case where $Y$ is the spectrum of a discrete valuation ring $A$.

The ring $C={\mathcal{O}}_{X,x}$ is then a localization of a finitely generated $A$-algebra, hence catenary (5.6.4), and by hypothesis the $C$-module $M_x/t{M}_x$ satisfies $(S_k)$ and is equidimensional (resp. one has $coprof(M_x/t{M}_x)\le k$); one therefore deduces from (5.12.2) (resp. from (0, 16.4.10)) that $M_x$ satisfies $(S_k)$ and is equidimensional (resp. that $coprof(M_x)\le k$), since $t$ is $M_x$-regular and belongs to the maximal ideal of $C$. Whence the conclusions, since $x^{\prime }$ is a generization of $x$ and $({\mathcal{F}}_{y^{\prime }}{)}_{x^{\prime }}={\mathcal{F}}_{x^{\prime }}$.

(12.1.1.3)

Let us pass to cases (vii) and (viii). One may evidently replace $Y$ by the integral sub-scheme having $\overline{y}$ as underlying space, and $X$ by $f^{-1}(\overline{y})$, without changing the fibres at $y$ and $y^{\prime }$; one may therefore suppose $A$ integral, with field of fractions $K=k(y)$. One knows ((4.5.11) and (4.7.8)) that there exists a finite extension $K^{\prime }$ of $K$ such that, for the $({\mathcal{O}}_X{\otimes }_A{K}^{\prime })$-Module ${\mathcal{F}}^{\prime }=\mathcal{F}{\otimes }_A{K}^{\prime }$, the associated prime cycles are geometrically irreducible and the geometric lengths of ${\mathcal{F}}^{\prime }$ at the maximal points $z^{\prime }$ of its support are respectively equal to the lengths of ${\mathcal{F}}_{z^{\prime }}^{\prime }$ at these points; it therefore amounts to the same to say that $\mathcal{F}$ is geometrically reduced (resp. geometrically pointwise integral) at a point $x^{\prime }$ of $X_{y^{\prime }}=X{\otimes }_{A}K$, or to say that ${\mathcal{F}}^{\prime }$ is reduced (resp. integral) at the points of $X{\otimes }_A{K}^{\prime }$ above $x^{\prime }$, taking (4.2.7) into account. Let $A^{\prime }$ be a finite $A$-algebra of which $K^{\prime }$ is the field of fractions; it follows from (4.7.11) that at every point of $X{\otimes }_A{A}^{\prime }$ above $x$, $\mathcal{F}{\otimes }_A{A}^{\prime }$ has property (vii) (resp. (viii)). Replacing $A$ by $A^{\prime }$ and $X$ by $X{\otimes }_A{A}^{\prime }$, one sees that one may confine oneself to proving that ${\mathcal{F}}_x$ is reduced (resp. integral) at every generization $x^{\prime }$ of $x$. One then proceeds as in (12.1.1.1), this time using (4.7.11), and one is once again reduced to the case where $A$ is a discrete valuation ring, $y$ being the closed point and $y^{\prime }$ the generic point of $Y$, with the uniformizer $t$ of $A$ being an $\mathcal{F}$-regular element. Since by hypothesis $\mathcal{F}/t\mathcal{F}$ is reduced (resp. integral) at the point $x$, it follows from (3.4.6) (resp. (3.4.5)) that $\mathcal{F}$ is reduced (resp. integral) at the point $x$, hence also in a neighbourhood of $x$, and in particular at the point $x^{\prime }$, which completes the proof in cases (vii) and (viii).

(12.1.1.4)

It remains to examine cases (i) and (ii). Replacing $Y$ by the integral sub-scheme having $\overline{y}$ as underlying space, one may confine oneself to the case where $Y$ is irreducible and $y$ its generic point. Let $z^{\prime }$ be a generic point of an associated prime cycle of ${\mathcal{F}}_{y^{\prime }}$

containing $x^{\prime }$, and let $Z^{\prime }$ be the closure of $z^{\prime }$ in $X$, so that one has $x\in Z^{\prime }$. To treat case (i), it will suffice to prove:

Proposition (12.1.1.5).

Let $Y$ be a locally Noetherian prescheme, $f:X\to Y$ a morphism locally of finite type, $\mathcal{F}$ a coherent ${\mathcal{O}}_X$-Module that is $f$-flat; for every $y\in Y$, set ${\mathcal{F}}_y=\mathcal{F}{\otimes }_{{\mathcal{O}}_Y}k(y)$. Let $z^{\prime }$ be a point of $X$, $y^{\prime }=f(z^{\prime })$, and suppose that $z^{\prime }\in Ass({\mathcal{F}}_{y^{\prime }})$. Let $Z^{\prime }$ (resp. $Y^{\prime }$) be the reduced sub-prescheme of $X$ (resp. $Y$) having as underlying space the closure of {z'} in $X$ (resp. of {y'} in $Y$). Then, for every $y\in f(Z^{\prime })$, the dimensions of all the irreducible components of $Z_y^{\prime }$ are equal to $\dim (Z_{y^{\prime }}^{\prime })$ (the dimension of the closure of $z^{\prime }$ in $X_{y^{\prime }}$) (which we shall express later (13.2.2) by saying that the restriction $Z^{\prime }\to Y^{\prime }$ of $f$ is an equidimensional morphism); moreover, at every maximal point $z$ of $Z_y^{\prime }$, one has $z\in Ass({\mathcal{F}}_y)$.

For this, we shall reduce to the case where $Y$ is the spectrum of a discrete valuation ring. Take a spectrum of a discrete valuation ring Y_1 and a morphism $h:Y_1\to X$ such that $h(y_1)=z$, $h(y_1^{\prime })=z^{\prime }$, where $y_1$ and $y_1^{\prime }$ are the closed point and the generic point of Y_1 respectively (II, 7.1.7). Let $g=f\circ h:Y_1\to Y$, $X_1=X{\times }_Y{Y}_1$, and let $f_1:X_1\to Y_1$, $g_1:X_1\to X$ be the canonical projections; there is a Y_1-section $h_1:Y_1\to X_1$ such that $h=g_1\circ h_1$ (I, 3.3.14). If $Z_1^{\prime }=Z_{y^{\prime }}^{\prime }{\otimes }_{k(y^{\prime })}k(y_1^{\prime })$, one knows (4.2.6) that the irreducible components of $Z_1^{\prime }$ dominate $Z_{y^{\prime }}^{\prime }$; let $z_1^{\prime }$ be the generic point of one of these components which contains $h_1(y_1^{\prime })$, so that $f_1(z_1^{\prime })=y_1^{\prime }$ and $g_1(z_1^{\prime })=z^{\prime }$. Since $h_1(y_1)$ is a specialization of $h_1(y_1^{\prime })$, it is a fortiori a specialization of $z_1^{\prime }$; let $z_1$ be a generic point of one of the irreducible components of $\overline{z_1^{\prime }}\cap (X_1{)}_{y_1}$ containing $h_1(y_1)$. Then $g_1(z_1)$ is a specialization of $g_1(z_1^{\prime })=z^{\prime }$ in $X$, and $z=h(y_1)=g_1(h_1(y_1))$ is a specialization of $g_1(z_1)$; but since $f(g_1(z_1))=y$ and $z$ is a generic point of $Z_y^{\prime }$, one necessarily has $z=g_1(z_1)$.

Suppose now that one has proved that, if one sets ${\mathcal{F}}_1=\mathcal{F}{\otimes }_{{\mathcal{O}}_Y}{\mathcal{O}}_{Y_1}$, one has $z_1\in Ass(({\mathcal{F}}_1{)}_{y_1})$; it will follow from (4.2.7) that $z\in Ass({\mathcal{F}}_y)$; moreover, the dimensions of $\overline{z_1}\cap (X_1{)}_{y_1}$ and of $\overline{z}\cap X_y$ are equal, and likewise the dimensions of $\overline{z_1^{\prime }}\cap (X_1{)}_{y_1^{\prime }}$ and of $\overline{z^{\prime }}\cap X_{y^{\prime }}$ (4.2.7); one has therefore indeed reduced, as announced, to the case where $Y$ is the spectrum of a discrete valuation ring $A$, $y$ and $y^{\prime }$ being its closed point and its generic point respectively.

This being so, since $z_1\in Ass(({\mathcal{F}}_1{)}_{y_1})$, one has a fortiori $z^{\prime }\in Ass(\mathcal{F})$. If $t$ is a uniformizer of $A$, $t$ is $\mathcal{F}$-regular by flatness, hence (3.4.3) one has $z\in Ass(\mathcal{F}/t\mathcal{F})=Ass({\mathcal{F}}_y)$. As for the assertion concerning dimensions, it follows from (7.1.13), applied to a closed sub-prescheme of $X$ having $Z^{\prime }$ as underlying space.

Let us tackle finally case (ii) of (12.1.1). With the same notation, one must prove that if all the associated prime cycles of ${\mathcal{F}}_y$ containing $x$ have dimension $k$, then the same is true of all the associated prime cycles of ${\mathcal{F}}_{y^{\prime }}$ containing $x^{\prime }$. Now we have just seen that every associated prime cycle of ${\mathcal{F}}_{y^{\prime }}$ containing $x^{\prime }$ has the same dimension as one of the associated prime cycles of ${\mathcal{F}}_y$ containing $x$; this therefore proves (ii).

Remarks (12.1.2).

(i) Under the conditions of (12.1.1.5) with $\mathcal{F}={\mathcal{O}}_X$ (so that $f$ is flat), one cannot in general assert that the restriction $Z^{\prime }\to Y^{\prime }$ of $f$ is a flat morphism, nor even an open morphism. This is what the example (6.5.5, (ii)) shows,

where one takes for $Z^{\prime }$ one of the two irreducible components of $X=\mathrm{Spec}(B)$. It is immediate that this restriction morphism is not open at the points of $Z^{\prime }$ above the "double point" of $Y^{\prime }=Y$.

(ii) In the hypotheses of (12.1.1.5), with $\mathcal{F}={\mathcal{O}}_X$, one cannot weaken the condition "$f$ is flat" to "$f$ is universally open" (cf. (2.4.6)), as we shall see later from an example (14.4.10, (i)).

Corollary (12.1.3).

Under the hypotheses of (12.1.1), the function $x\mapsto coprof(({\mathcal{F}}_{f(x)}{)}_x)$ is upper semi-continuous in $X$ and the function $x\mapsto pro{f}_x({\mathcal{F}}_x)$ (10.8.1) is lower semi-continuous in $X$.

The first assertion is none other than an equivalent formulation of (12.1.1, (v)). To prove the second, one may first, taking (10.8.7) and (10.8.8) into account, reduce as in (12.1.1) to the case where $Y$ is the spectrum of a discrete valuation ring $A$ with closed point $y$ and generic point $y^{\prime }$, with $X=\mathrm{Spec}(B)$ affine, $\mathcal{F}=\tilde{M}$, where $M$ is a $B$-module of finite type. Since $\mathcal{F}$ is $f$-flat, every irreducible component of $Supp(\mathcal{F})$ that contains $x\in X_y$ dominates $Y$ (2.3.4), in other words its generic point $z^{\prime }$ belongs to $X_{y^{\prime }}$; the irreducible components $Z$ of $Supp(\mathcal{F})$ that contain a generization $x^{\prime }$ of $x$ belonging to $X_{y^{\prime }}$ are therefore exactly those that contain $x$, and moreover, by (12.1.1.5), one has $\dim (Z_{y^{\prime }})=\dim (Z_y)$, whence, if $T=Supp(\mathcal{F})$, ${\dim }_x(T_y)={\dim }_{x^{\prime }}(T_{y^{\prime }})$. In addition, for a uniformizer $t$ of $A$, one saw in (12.1.1, (v)) that one has $coprof(M_x)=coprof(M_x/t{M}_x)$, and since on the other hand $coprof(M_{x^{\prime }})\le coprof(M_x)$ (6.11.5), one sees that one has $copro{f}_{x^{\prime }}({\mathcal{F}}_{y^{\prime }})\le copro{f}_x({\mathcal{F}}_y)$; the relation $pro{f}_{x^{\prime }}({\mathcal{F}}_{y^{\prime }})\ge pro{f}_x({\mathcal{F}}_y)$ then follows from (10.8.7).

Remark (12.1.4).

One also deduces from (12.1.1, (i)) that the function $x\mapsto {\dim }_x(X_{f(x)})$ is upper semi-continuous in $X$, but we shall see later (13.1.3) that this property is true even without supposing $f$ flat.

Corollary (12.1.5).

Let $Y$ be a locally Noetherian prescheme, $f:X\to Y$ a morphism locally of finite type, $\mathcal{F}$ a coherent ${\mathcal{O}}_X$-Module that is $f$-flat, $\mathcal{G}$ a coherent ${\mathcal{O}}_Y$-Module, $x$ a point of $X$, $y=f(x)$. Suppose that $\mathcal{G}$ has property $(S_k)$ at the point $y$, and that ${\mathcal{F}}_y$ has property $(S_k)$ at the point $x$ and is equidimensional at the point $x$. Then:

(i) $\mathcal{F}{\otimes }_{{\mathcal{O}}_Y}\mathcal{G}$ possesses property $(S_k)$ at the point $x$.

(ii) If moreover $Y$ is locally immersible in a regular scheme, or is an excellent prescheme (7.8.5), there exists a neighbourhood of $x$ in $X$ such that $\mathcal{F}{\otimes }_{{\mathcal{O}}_Y}\mathcal{G}$ has property $(S_k)$ in this neighbourhood.

Indeed, by (12.1.1, (iv)), there exists an open neighbourhood $V$ of $x$ such that for every $x^{\prime }\in V$, ${\mathcal{F}}_{f(x^{\prime })}$ satisfies $(S_k)$ at the point $x^{\prime }$. On the other hand, $\mathcal{G}$ satisfies $(S_k)$ at every point of $Y^{\prime }=\mathrm{Spec}({\mathcal{O}}_y)$ (5.7.2). Replacing $Y$ by $Y^{\prime }$ and $X$ by $V{\times }_Y{Y}^{\prime }$, one is reduced (taking (I, 3.6.5) into account) to the case where $\mathcal{G}$ satisfies $(S_k)$ in $Y$ entirely and where, for every $y\in f(X)$, ${\mathcal{F}}_y$ possesses property $(S_k)$ at every point of $f^{-1}(y)$; since $\mathcal{F}$ is $f$-flat, one then knows (6.4.1) that $\mathcal{F}{\otimes }_{{\mathcal{O}}_Y}\mathcal{G}$ possesses property $(S_k)$ in $X$, which proves (i). To prove (ii), it suffices to observe that, under the hypotheses made, $\mathcal{G}$ possesses

property $(S_k)$ in a neighbourhood $U$ of $y$ in $Y$ ((6.11.2) and (7.8.3, (iv))); one then concludes in the same way, replacing this time $Y$ by $U$ and $X$ by $V{\times }_{Y}U$.

Theorem (12.1.6).

Let $f:X\to Y$ be a flat morphism locally of finite presentation, $k$ an integer. The following subsets of $X$ are open:

(i) The set of points $x\in X$ such that $X_{f(x)}$ satisfies property $(S_k)$ at the point $x$.

(ii) The set of $x\in X$ such that $X_{f(x)}$ satisfies geometric property $(R_k)$ at the point $x$, is equidimensional at the point $x$, and moreover $x$ belongs to no embedded associated prime cycle of $X_{f(x)}$.

(iii) The set of $x\in X$ such that $X_{f(x)}$ is geometrically regular (i.e. smooth) at the point $x$.

(iv) The set of $x\in X$ such that $X_{f(x)}$ is geometrically normal at the point $x$.

Steps A) and B) of (12.0.2) are carried out here as in (12.1.1); for step A), one uses (6.7.8), as well as (4.2.7) for (ii); for step B), one uses (9.9.2) and (9.9.4). It remains to examine step C) in each case.

(i) As in (12.1.1.2), one reduces (using (6.7.8)) to the case where $Y$ is the spectrum of a discrete valuation ring $A$, $X=\mathrm{Spec}(B)$, where $B$ is an $A$-algebra of finite type, $x$, $x^{\prime }$ two points of $X$ such that $f(x)=y$ is the closed point and $y^{\prime }=f(x^{\prime })$ the generic point of $Y$, with $x^{\prime }$ moreover a generization of $x$. Since the task is to prove that $X$ satisfies $(S_k)$ at the point $x^{\prime }$, one may replace $X$ by $\mathrm{Spec}({\mathcal{O}}_{X,x^{\prime }})$, that is, suppose the ring $B$ local, the homomorphism $A\to B$ local, and $B$ essentially of finite type over $A$ (1.3.8). Then, by hypothesis, if $t$ is a uniformizer of $A$, $t$ is a $B$-regular element by flatness, and $B/tB$ satisfies $(S_k)$. But since $A$ is a universally catenary ring (5.6.4), $B$ is catenary, hence, by (5.12.4), it follows that $B$ satisfies $(S_k)$, which completes the proof in case (i).

(iii) Here step C) is unnecessary; since $f$ is flat, one knows in effect (6.8.7) that when $Y$ is locally Noetherian and $f$ locally of finite type, the set of $x\in X$ such that $X_{f(x)}$ is geometrically regular at the point $x$ is open in $X$.

(ii) Reasoning as in (12.1.1.3), to prove that the property considered is stable under generization, one may first, by considering an arbitrary finite extension $K^{\prime }$ of $k(y)$, and taking account of definition (6.7.6) of geometric property $(R_k)$, as well as of the invariance under base-field change of the two other properties figuring in (ii), replace in (ii) the geometric property $(R_k)$ by property $(R_k)$. Proceeding then as in (i), one reduces to the case where $Y$ is the spectrum of a discrete valuation ring $A$, and since $A$ is catenary, it suffices to apply (5.12.5) to conclude.

(iv) The set of points $x\in X$ such that $X_{f(x)}$ is geometrically normal at the point $x$ is contained in the set of $x\in X$ such that $X_{f(x)}$ is geometrically pointwise integral and satisfies (S_2) at the point $x$, and the latter is open in $X$ by virtue of (12.1.1, (viii)). One may therefore already suppose that, for every $x\in X$, $X_{f(x)}$ is geometrically pointwise integral and satisfies (S_2) at the point $x$, and a fortiori it is equidimensional. On the other hand, by virtue of Serre's criterion (5.8.6), to say that $X_{f(x)}$ is geometrically normal at the point $x$ means that $X_{f(x)}$ satisfies (S_2) and geometric property (R_1) at $x$;

but by virtue of (ii) and the preceding remarks, this set is the intersection of two open sets of $X$, hence is open in $X$. Q.E.D.

Corollary (12.1.7).

Let $f:X\to Y$ be a morphism locally of finite presentation. Then the set of points $x\in X$ where $f$ possesses one of the following properties (6.8.1):

(i) satisfies property $(S_n)$;

(ii) is of codepth $\le n$;

(iii) is Cohen-Macaulay;

(iv) is regular (or smooth, which amounts to the same);

(v) is normal;

(vi) is reduced;

is open in $X$.

Indeed, it follows from (11.3.1) that the set of $x\in X$ where $f$ is flat is open. One may therefore confine oneself to the case where $f$ is flat, and then the corollary follows from (12.1.1, (iv), (vi), and (vii)) and from (12.1.6, (i), (ii), and (iv)).

Remarks (12.1.8).

(i) In (12.1.6, (ii)), one cannot suppress the hypothesis that $x$ belongs to no embedded associated prime cycle of $X_{f(x)}$. This is what the example (5.12.3) shows, where one takes $X=\mathrm{Spec}(A)$, $Y$ being the spectrum of the local ring A_0 of k[T] corresponding to the prime ideal (T) and the morphism $X\to Y$ corresponding to the injective homomorphism $A_0\to A$, deduced by localization and passage to the quotient from the injection $k[T]\to k[T,U,V]$; this morphism is flat since $A$ is a torsion-free A_0-module $(0_I,\mathrm{6.3.4})$. Here the fibre $X_y$ at the closed point $y$ of $Y$, equal to $\mathrm{Spec}(A/tA)$, is irreducible, of dimension 1, and satisfies the geometric condition (R_0), since the local ring at its generic point is a field. By contrast, at the generic point $y^{\prime }$ of $Y$, the fibre $X_{y^{\prime }}$ has two irreducible components of respective dimensions 0 and 1, and the one of dimension 0 is not reduced, so $X_{y^{\prime }}$ does not satisfy condition (R_0).

(ii) In (12.1.6, (ii)), neither can one suppress the hypothesis that $X_{f(x)}$ is equidimensional at the point $x$. One sees this here on the example (5.12.6) with $X=\mathrm{Spec}(A)$, $Y=\mathrm{Spec}(A_0)$, where A_0 is defined as in (i), the morphism $X\to Y$ coming again from the injection $k[T]\to k[T,U,V,W]$ by localization and passage to the quotient, and being flat since $A$ is a torsion-free A_0-module $(0_I,\mathrm{6.3.4})$. The fibre $X_y$ at the closed point $y$ of $Y$ is reduced (hence satisfies (S_1)) and satisfies (R_1), but has two irreducible components of dimensions 2 and 1, while the fibre $X_{y^{\prime }}$ at the generic point $y^{\prime }$ of $Y$ does not satisfy [condition (R_1)].

12.2. Local and global properties of the fibres of a proper, flat morphism of finite presentation

Theorem (12.2.1).

Let $f:X\to Y$ be a proper morphism of finite presentation, $\mathcal{F}$ an ${\mathcal{O}}_X$-Module that is $f$-flat and of finite presentation, $\Phi$ a finite subset of $\mathbb{Z}\cup \pm \infty$, $k$ an integer. The following subsets of $Y$ are open:

(i) The set of $y\in Y$ such that the set of dimensions of the associated prime cycles of ${\mathcal{F}}_y$ is contained in $\Phi$.

(ii) The set of $y\in Y$ such that the set of dimensions of the irreducible components of $Supp({\mathcal{F}}_y)$ contains $\Phi$.

(iii) The set of $y\in Y$ such that all the associated prime cycles of ${\mathcal{F}}_y$ have the same dimension equal to $k$.

(iv) The set of $y\in Y$ such that ${\mathcal{F}}_y$ has no embedded associated prime cycle and that the set of dimensions of the irreducible components of $Supp({\mathcal{F}}_y)$ is equal to $\Phi$.

(v) The set of $y\in Y$ such that ${\mathcal{F}}_y$ has property $(S_k)$ and is equidimensional at every point of $X_y$.

(vi) The set of $y\in Y$ such that $coprof({\mathcal{F}}_y)\le k$.

(vii) The set of $y\in Y$ such that ${\mathcal{F}}_y$ is a Cohen-Macaulay ${\mathcal{O}}_X$-Module.

(viii) The set of $y\in Y$ such that ${\mathcal{F}}_y$ is geometrically reduced.

(ix) The set of $y\in Y$ such that ${\mathcal{F}}_y$ is geometrically pointwise integral at each point of $X_y$.

(x) The set of $y\in Y$ such that ${\mathcal{F}}_y$ is geometrically integral.

(xi) The set of $y\in Y$ such that ${\mathcal{F}}_y$ has no embedded associated prime cycle and that the sum $l(y)$ of the total multiplicities (4.7.12) of ${\mathcal{F}}_y$ at the maximal points of $Supp({\mathcal{F}}_y)$ is $\le k$.

With the exception of (ii), (iii), (iv), (x), and (xi), the properties $P(y)$ considered are of the following form: "for every $x\in X_y$, ${\mathcal{F}}_y$ has property $Q(x)$", where $Q(x)$ is one of properties (i) to (viii) of (12.1.1). It follows from (12.1.1) that the set $U$ of $x\in X$ such that $Q(x)$ is true is open, and the set $V$ of $y\in Y$ such that $P(y)$ is true is none other than the set $Y-f(X-U)$; in all these cases, the theorem is therefore already true on the sole hypothesis that the morphism $f$ is closed. For (iii), one applies (i) with $\Phi$ reduced to a single element. There remain cases (ii), (iv), and (xi) to examine separately ((x) deducing at once from (xi) and (4.7.14)), always applying the method described in (12.0.2).

(12.2.1.1)

Case (ii): Steps A) and B) of the proof proceed as in the beginning of (12.1.1); for step A), one uses (8.9.1), (11.2.6), and (4.2.7); for step B), one uses (9.5.5) applied to $Supp(\mathcal{F})$. It remains to show that if (supposing $Y$ Noetherian) the set of dimensions of the irreducible components of $Supp({\mathcal{F}}_y)$ contains $\Phi$, then the same is true of the set of dimensions of the irreducible components of $Supp({\mathcal{F}}_{y^{\prime }})$, for every generization $y^{\prime }$ of $y$ in $Y$. Let Y_1 be a spectrum of a discrete valuation ring such that, if $y_1$ and $y_1^{\prime }$ are the closed point and the generic point of Y_1, there is a morphism $h:Y_1\to Y$ with $h(y_1)=y$, $h(y_1^{\prime })=y^{\prime }$ (II, 7.1.7). Applying (4.2.7), one sees that one may replace $Y$ by Y_1 and $X$ by $X_1=X{\times }_Y{Y}_1$, in other words confine oneself to the case where $Y$ is the spectrum of a discrete valuation ring, $y$ its closed point and $y^{\prime }$ its generic point.

Using $(Er{r}_{III},30)$, one may confine oneself to the case where $Supp(\mathcal{F})=X$, so that $f$ is quasi-flat (2.3.3); the irreducible components $Z_i$ of $X$ then dominate $Y$ (2.3.4), in other words their generic points belong to $X_{y^{\prime }}$ and the $Z_i\cap X_{y^{\prime }}$ are the irreducible components of $X_{y^{\prime }}$. But every irreducible component of $X_y$ is contained in one of the $Z_i$, hence is an irreducible component of $(Z_i{)}_y$; now one knows (7.1.13) that the dimensions of all the irreducible components of $(Z_i{)}_y$ are equal to $\dim ((Z_i{)}_{y^{\prime }})$, which completes the proof of (ii).

(12.2.1.2)

Case (iv): The same reasoning as at the beginning shows, using (12.1.1, (iii)), that one may already suppose (replacing $Y$ by an open set of $Y$) that for every $y\in Y$, ${\mathcal{F}}_y$ has no embedded associated prime cycle. The assertion of case (iv) is then an immediate consequence of the assertions of cases (i) and (ii).

(12.2.1.3)

Case (xi): For step A) of the reasoning, one uses (8.9.1), (11.2.6), (8.10.5, (xii)) (to preserve the hypothesis that $f$ is proper), as well as (4.2.7), (4.5.6), and (4.7.9). For step B), one sees, as in case (iv), that one may suppose that for every $y\in Y$, ${\mathcal{F}}_y$ has no embedded associated prime cycle, and one applies (9.8.8), which proves that the function $z\mapsto l(z)$ is constructible. It therefore remains to see (supposing $Y$ Noetherian and $f$ proper) that for every generization $y^{\prime }$ of a point $y\in Y$, one has $l(y^{\prime })\le l(y)$. Reasoning as in (12.1.1.3), one sees (using (4.7.8) and (4.5.11)) that one may suppose that the irreducible components of $Supp({\mathcal{F}}_{y^{\prime }})$ are geometrically irreducible and that $l(y^{\prime })$ is the sum of the lengths of $({\mathcal{F}}_{y^{\prime }}{)}_{z_j^{\prime }}$ at the maximal points $z_j^{\prime }$ of $Supp({\mathcal{F}}_{y^{\prime }})$. Applying (4.2.7), (4.5.6), and (4.7.9) again, one reduces, as in (12.2.1.1), to the case where $Y$ is the spectrum of a discrete valuation ring, $y$ its closed point and $y^{\prime }$ its generic point. The fact that $l(y^{\prime })\le l(y)$ will then be a consequence of:

Lemma (12.2.1.4).

Let $Y$ be the spectrum of a discrete valuation ring, $y$ its closed point, $y^{\prime }$ its generic point, $f:X\to Y$ a proper morphism, $\mathcal{F}$ a coherent ${\mathcal{O}}_X$-Module that is $f$-flat, $z_i$ (resp. $z_j^{\prime }$) the maximal points of $Supp({\mathcal{F}}_y)$ (resp. $Supp({\mathcal{F}}_{y^{\prime }})$). Suppose that ${\mathcal{F}}_y$ has no embedded associated prime cycle. Then one has

  (12.2.1.4.1)         ∑_j long((ℱ_{y'})_{z'_j}) ≤ ∑_i long((ℱ_y)_{z_i}).

One has $Y=\mathrm{Spec}(A)$, where $A$ is a discrete valuation ring, of which we denote by $t$ a uniformizer, so that ${\mathcal{F}}_y=\mathcal{F}/t\mathcal{F}$. Since $\mathcal{F}$ is $f$-flat, the $z_j^{\prime }$ are also the maximal points of $Supp(\mathcal{F})$ (2.3.4); for every $z_i$, let us denote by $z_{ij}^{\prime }$ those of the $z_j^{\prime }$ that are generizations of $z_i$; it follows from (3.4.1.1) that one has

$$long(({\mathcal{F}}_y{)}_{z_i})\ge \sum _{j}long(({\mathcal{F}}_{y^{\prime }}{)}_{z_{ij}^{\prime }}),$$

whence on summing

  ∑_i long((ℱ_y)_{z_i}) ≥ ∑_{i, j} long((ℱ_{y'})_{z'_{ij}}).

The lemma will therefore be proved if we establish that for every $z_j^{\prime }$ there is at least one index $i$ such that $z_j^{\prime }$ is one of the $z_{ij}^{\prime }$. Now, since $f$ is proper (hence closed) and $f(z_j^{\prime })=y^{\prime }$, there exists $x\in X$ which is a specialization of $z_j^{\prime }$ and is such that $f(x)=y$, in other words $\overline{z_j^{\prime }}\cap X_y$ is non-empty. Since $t$ is $\mathcal{F}$-regular by flatness, one deduces from (3.4.3) that there is at least one point of $Ass({\mathcal{F}}_y)$ of which $z_j^{\prime }$ is a generization; but since the points of $Ass({\mathcal{F}}_y)$ are by hypothesis the $z_i$, this completes the proof.

Corollary (12.2.2).

Under the hypotheses of (12.2.1):

(i) The function $y\mapsto \dim (Supp({\mathcal{F}}_y))$ is continuous (hence locally constant) in $Y$.

(ii) The function $y\mapsto coprof({\mathcal{F}}_y)$ (5.7.1) is upper semi-continuous in $Y$.

(iii) The function $y\mapsto l(y)$ (12.2.1, (xi)) is upper semi-continuous in $Y$ when the ${\mathcal{F}}_y$ have no embedded associated prime cycle.

(iv) The function $y\mapsto pro{f}_{\ast }({\mathcal{F}}_y)$ (10.8.1) is lower semi-continuous in $Y$.

(i) It follows from (12.2.1, (i)) applied to $\Phi$ equal to the smallest interval of $\mathbb{N}$ containing the dimensions of the associated prime cycles of ${\mathcal{F}}_y$, that $z\mapsto \dim (Supp({\mathcal{F}}_z))$ is upper semi-continuous at the point $y$; it follows on the other hand from (12.2.1, (ii)) applied to $\Phi$ equal to the set of dimensions of the irreducible components of $Supp({\mathcal{F}}_y)$, that this same function is lower semi-continuous at the point $y$, whence the conclusion. Assertions (ii) and (iii) are nothing but other formulations of (12.2.1, (vi) and (xi)). Finally, by (12.1.3), the set of $x\in X$ such that $pro{f}_x({\mathcal{F}}_x)\ge k$ is open, and the conclusion of (iv) follows by the same reasoning as at the beginning of (12.2.1).

Remarks (12.2.3).

(i) One will observe that for (12.2.1, (ii)), one may dispense with the hypothesis that $f$ is proper.

(ii) In (12.2.1, (ii)), one cannot replace the condition that the set $E(y)$ of dimensions of the irreducible components of $Supp({\mathcal{F}}_y)$ contain $\Phi$, by the condition that $E(y)$ be equal to $\Phi$. Indeed, let $k$ be a field, and consider the projective space of dimension 3, $P=P_k^3=\mathrm{Proj}(C)$, where $C=k[t_0,t_1,t_2,t_3]$ (with $t_i$ indeterminates); in $P$, let X_0 be the closed sub-scheme "union of the line X_1 defined by $t_1=t_2=0$ and of the plane X_2 defined by $t_2-t_3=0$", which describes in geometric terms the fact that X_0 is equal to $\mathrm{Proj}(C/\mathfrak{a})$, where the graded ideal $\mathfrak{a}$ is equal to $\mathfrak{bc}$, with $\mathfrak{b}=C{t}_1+C{t}_2$, $\mathfrak{c}=C(t_2-t_3)$; consider the morphism $f_0:X_0\to X_1$ which, in geometric terms, is the "projection of X_0 onto X_1 from the line at infinity $D$ defined by $t_0-t_2=0$"; in algebraic form, $f_0$ corresponds to the homomorphism of graded algebras $k[t_0,t_1]\to k[t_0,t_1,t_2,t_3]/\mathfrak{a}$ obtained by passage to the quotient from the canonical injection $k[t_0,t_1]\to k[t_0,t_1,t_2,t_3]$. It is clear that $f_0$ is a projective morphism, hence proper; so is its restriction $f:X\to Y$, where $Y=D_{+}(t_0)$ and $X=f_0^{-1}(Y)$; to see that $f$ is flat, it suffices to remark that $k[t_1]$ is a principal ring and the ring $C^{\prime }=k[t_1,t_2,t_3]/(t_2-t_3)(t_1)+(t_2)$ a torsion-free $k[t_1]$-module $(0_I,\mathrm{6.3.4})$. For every $y\in Y$ distinct from the point $y_0$ defined by $t_2=0$, $f^{-1}(y)$ then has two irreducible components, of dimensions 0 and 1, while $f^{-1}(y_0)$ has only one irreducible component of dimension 1.

(iii) In (12.2.1, (i)), one cannot either replace the condition that the set $E^{\prime }(y)\supset E(y)$ of dimensions of the associated prime cycles of ${\mathcal{F}}_y$ be contained in $\Phi$ by the condition that it be equal to $\Phi$. Indeed, let $k$ be a field, $A$ the polynomial ring k[t], $B$ the sub-ring $k[t^4,t^{3}u,t{u}^3,u^4]$ of the polynomial ring k[t, u] (with $t$, $u$ indeterminates); $B$ is not integrally closed, the element $t^2{u}^2$ belonging to its integral closure but not being in $B$; if $\mathfrak{m}$ is the maximal ideal of $B$, generated by $t^4$, $t^{3}u$, $t{u}^3$, and $u^4$, $\mathfrak{m}$ is an embedded associated prime ideal of $A/tA$. Set $Y=\mathrm{Spec}(A)$, $X=\mathrm{Spec}(B)$, and let $f:X\to Y$ be the morphism corresponding to the homomorphism $A\to B$ which transforms $t$ into $t^4$. Since this homomorphism makes $B$ a torsion-free $A$-module, $f$ is flat $(0_I,\mathrm{6.3.4})$. If $y$ is the point of $Y$ corresponding to the maximal ideal tA, the prescheme $f^{-1}(y)$ is irreducible and of dimension 1, but admits an embedded associated prime cycle of dimension 0; on the contrary, if $y^{\prime }$ is the generic point of $Y$, $f^{-1}(y^{\prime })$ has no embedded associated prime cycle, being the spectrum of an integral $k(y^{\prime })$-algebra of dimension 1. In this example, $f$ is an affine morphism that is not proper; one can immediately deduce from it an analogous example where $f$ is proper and flat by considering $X$ as a dense open set of a projective scheme over $Y$ ((II, 5.3.4 and 5.3.2)), or by proceeding directly as in Remark (ii).

Remarks (ii) and (iii) show the necessity of including in (12.2.1, (iv)) the condition that ${\mathcal{F}}_y$ have no embedded associated prime cycle.

(iv) In (12.2.1, (xi)), one cannot suppress the hypothesis that $f$ is proper. Indeed, let $k$ be a field, $Y$ the "affine line", spectrum of the polynomial ring k[t] ($t$ indeterminate), $X$ the sum prescheme of $Y$ and of the complementary open set of the closed point $y$ of $Y$ defined by $t=0$. It is clear that the morphism $f:X\to Y$ which is equal to the canonical injections on each of the components of $X$ is flat and that if one takes $\mathcal{F}={\mathcal{O}}_X$, ${\mathcal{F}}_z$ has no embedded associated prime cycle for any $z\in Y$. However, one sees at once that one has $l(y)=1$ and $l(z)=2$ for every $z\ne y$.

(v) In (12.2.1, (xi)), neither can one suppress the hypothesis that ${\mathcal{F}}_y$ is without embedded associated prime cycle. This is what the example of Remark (ii) shows: one sees at once in effect that one has, with $\mathcal{F}={\mathcal{O}}_X$, $l(y_0)=1$ and $l(y)=2$ for $y\ne y_0$ in $Y$.

(vi) We do not know whether in (12.2.1, (xi)) one may replace the hypothesis that $\mathcal{F}$ is $f$-flat by the hypothesis that $Supp(\mathcal{F})=X$ and that $f$ is universally open. Taking up the proof of (12.2.1.4) again, one sees (also using (14.3.6)) that one would have to resolve the following question: let $A$ be a Noetherian local ring, $M$ an $A$-module

of finite type, $t$ an element of the maximal ideal $\mathfrak{m}$ of $A$ that is contained in no minimal prime ideal of $A$; if there exists one of these minimal prime ideals $\mathfrak{p}$ such that $\dim (A/\mathfrak{p})=1$, is it true that $\mathfrak{m}$ is an associated prime ideal of $M/tM$ ($t$ not being supposed $M$-regular)?

One will note however that one cannot, in (12.2.1, (xi)), suppress purely and simply the hypothesis that $\mathcal{F}$ is $f$-flat. One will take here $P$ as in Remark (ii), then in $P$ the closed sub-scheme X_0 union of the three lines X_1, X_2, X_3 defined respectively by $t_1=t_3=0$, $t_1-t_0=t_3=0$, $t_2=t_3=0$; one defines the projection $f_0$ of X_0 onto X_1 as before, and its restriction $f:X\to Y$, where $Y=D_{+}(t_0)$ is the affine line and $X=f_0^{-1}(Y)$; $f$ is proper but not flat, and if one takes $\mathcal{F}={\mathcal{O}}_X$, ${\mathcal{F}}_y$ has no embedded associated prime cycles for any $y\in Y$. But one has $l(y_0)=1$ and $l(y)=2$ for $y\ne y_0$ in $Y$.

Theorem (12.2.4).

Let $f:X\to Y$ be a proper, flat morphism of finite presentation, $k$ an integer $\ge 1$. The following subsets of $Y$ are open:

(i) The set of $y\in Y$ such that $X_y$ possesses property $(S_k)$.

(ii) The set of $y\in Y$ such that $X_y$ satisfies geometric property $(R_k)$, is equidimensional at every point, and has no embedded associated prime cycle.

(iii) The set of $y\in Y$ such that $X_y$ is geometrically regular (i.e. smooth over $k(y)$).

(iv) The set of $y\in Y$ such that $X_y$ is geometrically normal.

(v) The set of $y\in Y$ such that $X_y$ is geometrically reduced.

(vi) The set of $y\in Y$ such that $X_y$ is geometrically reduced and that the geometric number of connected components of $X_y$ is equal to $k$.

(vii) The set of $y\in Y$ such that $X_y$ is geometrically pointwise integral (4.6.9).

(viii) The set of $y\in Y$ such that $X_y$ is geometrically integral.

(ix) The set of $y\in Y$ such that $X_y$ has no embedded associated prime cycle and that the total multiplicity (4.7.4) of $X_y$ over $k(y)$ is $\le k$.

Except for (vi), these assertions are special cases of assertions of (12.2.1), or are deduced from assertions of (12.1.6) as at the beginning of the proof of (12.2.1). For (vi), one reduces as always (taking into account the invariance under base change of the geometric number of connected components (4.5.6)) to the case where $Y$ is Noetherian. The set $U$ of $y\in Y$ such that $X_y$ is geometrically reduced is open in $Y$ by virtue of (v); it then follows from (III, 7.8.7 and 7.8.6) that for every $y\in U$, there is a neighbourhood $V\subset U$ of $y$ and an integer $m$ such that, for every $z\in V$, $\Gamma (X_z,{\mathcal{O}}_{X_z})$ is isomorphic to $(k(z){)}^m$; but by virtue of (III, 4.3.4), $m$ is then the geometric number of connected components of $X_z$, whence the conclusion. One will give another proof of (vi) in (15.5.9).

12.3. Local cohomological properties of the fibres of a flat morphism locally of finite presentation

Lemma (12.3.1).

Let $A$ be a ring, $B$ an $A$-algebra of finite presentation that is a flat $A$-module, $M$ a $B$-module of finite presentation that is a flat $A$-module. Then the following conditions are equivalent:

a) $M$ is a projective $B$-module.

b) $M$ is a flat $B$-module.

c) For every $y\in \mathrm{Spec}(A)$, $M{\otimes }_{A}k(y)$ is a projective $(B{\otimes }_{A}k(y))$-module.

The equivalence of a) and b) follows from Bourbaki, Alg. comm., chap. II, §5, n° 2, cor. 2 of th. 1. Since a) implies c) trivially, it remains to prove that c) implies b), which follows from the fibrewise flatness criterion (11.3.10), applied with $g=h$, $f=1_X$.

Proposition (12.3.2).

Let $A$ be a ring, $B$ an $A$-algebra of finite presentation that is a flat $A$-module, $M$ a $B$-module of finite presentation that is a flat $A$-module. Then:

(i) There exists a left resolution of $M$ by free $B$-modules of finite type.

(ii) One has

  (12.3.2.1)    dim. proj_B(M) = sup_{y ∈ Spec(A)} dim. proj_{B ⊗_A k(y)}(M ⊗_A k(y)) = Tor.dim_B(M)

where $Tor.{\dim }_B(M)$ is the smallest integer $i$ such that $To{r}_j^B(M,N)=0$ for every $j>i$ and every $B$-module $N$ (and $+\infty$ if no such integer $i$ exists).

(i) By virtue of (8.9.1), there exists a Noetherian sub-ring A_0 of $A$, an A_0-algebra of finite type B_0 and a B_0-module of finite type M_0 such that $B$ is isomorphic to $B_0{\otimes }_{A_0}A$ and $M$ to $M_0{\otimes }_{A_0}A$. Moreover (11.2.7), one may suppose that B_0 and M_0 are flat A_0-modules. There then exists a left resolution $(L_0{)}_{\bullet }$ of M_0 formed of free B_0-modules of finite type, and since M_0 and the $L_{0,i}$ are flat A_0-modules, $(L_0{)}_{\bullet }{\otimes }_{A_0}A$ is a left resolution of $M$ formed of free $B$-modules of finite type (2.1.10).

(ii) By virtue of (0, 17.2.2, (ii)), one has Tor.dim_B(M) ≤ dim. proj_B(M), and the definition of the projective dimension immediately shows that, for every $y\in \mathrm{Spec}(A)$, one has dim. proj_{B ⊗_A k(y)}(M ⊗_A k(y)) ≤ dim. proj_B(M). To prove the reverse inequalities, consider a left resolution $(L_i)$ of $M$ by free $B$-modules of finite type, and suppose that $Tor.{\dim }_B(M)=n$ (resp. $\dim .pro{j}_{B{\otimes }_{A}k(y)}(M{\otimes }_{A}k(y))\le n$ for every $y\in \mathrm{Spec}(A)$). Then $R=Im(L_n\to L_{n-1})=Ker(L_{n-1}\to L_{n-2})$ is a $B$-module of finite type that is also a flat $A$-module, by virtue of the hypothesis on $M$ and $B$ and of (2.1.10). In addition, one has $To{r}_{j+1}^B(M,N)=To{r}_j^B(R,N)$ for every $B$-module $N$ (M, V, 7). The hypothesis $Tor.{\dim }_B(M)=n$ therefore entails $To{r}_1^B(R,N)=0$ for every $B$-module $N$, that is to say that $R$ is a flat $B$-module, hence projective by virtue of (12.3.1); this establishes that $\dim .pro{j}_B(M)\le n$. The hypothesis $\dim .pro{j}_{B{\otimes }_{A}k(y)}(M{\otimes }_{A}k(y))\le n$ for every $y\in \mathrm{Spec}(A)$ entails on the other hand, by tensorization with $k(y)$, that in each of the sequences (exact by virtue of the flatness over $A$ of $M$, of the $L_i$, and of $R$ (2.1.10))

  0 → R ⊗_A k(y) → L_{n−1} ⊗_A k(y) → ⋯ → L_0 ⊗_A k(y) → M ⊗_A k(y) → 0,

$R{\otimes }_{A}k(y)$ is a projective $(B{\otimes }_{A}k(y))$-module (for $y\in \mathrm{Spec}(A)$). One concludes once again from (12.3.1) that $R$ is a projective $B$-module, hence $\dim .pro{j}_B(M)\le n$.

Proposition (12.3.3).

Let $f:X\to S$ be a morphism locally of finite presentation, ${\mathcal{L}}_{\bullet }$ a complex formed of quasi-coherent ${\mathcal{O}}_X$-Modules of finite presentation; for every $s\in S$, let $({\mathcal{L}}_{\bullet }{)}_s$ be the complex ${\mathcal{L}}_{\bullet }{\otimes }_{{\mathcal{O}}_S}k(s)$ of ${\mathcal{O}}_{X_s}$-Modules of finite type. Suppose that ${\mathcal{L}}_{n-1}$ is $f$-flat. Then the set $U$ of $x\in X$ such that $({\mathcal{H}}_n(({\mathcal{L}}_{\bullet }{)}_{f(x)}){)}_x=0$ is open in $X$. If moreover ${\mathcal{L}}_n$ is $f$-flat, then one has ${\mathcal{H}}_n({\mathcal{L}}_{\bullet })|U=0$.

One may evidently confine oneself to the case where $n=1$ and where the complex ${\mathcal{L}}_{\bullet }$ reduces to $0\to {\mathcal{L}}_2\to {\mathcal{L}}_1\to {\mathcal{L}}_0\to 0$. One may first reduce to the case where $S=\mathrm{Spec}(A)$ and $X$

are affine, then to the case where $S$ is Noetherian; indeed, by (8.9.1) and (8.5.2), one knows that there exist a Noetherian prescheme $S^{\prime }=\mathrm{Spec}(A^{\prime })$, where $A^{\prime }$ is a sub-ring of $A$, a morphism of finite type $f^{\prime }:X^{\prime }\to S^{\prime }$, and three coherent ${\mathcal{O}}_{X^{\prime }}$-Modules ${\mathcal{L}}_i^{\prime }$ ($0\le i\le 2$) such that $X=X^{\prime }{\times }_{S^{\prime }}S$, $f=f_{(S)}^{\prime }$, ${\mathcal{L}}_i={\mathcal{L}}_i^{\prime }{\otimes }_{S^{\prime }}S$, as well as two homomorphisms $u^{\prime }$, $v^{\prime }$ such that $u=u^{\prime }\otimes 1$, $v=v^{\prime }\otimes 1$, and $v^{\prime }\circ u^{\prime }=0$. One may moreover suppose that ${\mathcal{L}}_0^{\prime }$ is $f^{\prime }$-flat (11.2.7), and that ${\mathcal{L}}_1^{\prime }$ is $f^{\prime }$-flat when ${\mathcal{L}}_1$ is supposed $f$-flat. By faithful flatness, the hypothesis $({\mathcal{H}}_1(({\mathcal{L}}_{\bullet }{)}_{f(x)}){)}_x=0$ is equivalent to $({\mathcal{H}}_1(({\mathcal{L}}_{\bullet }^{\prime }{)}_{f^{\prime }(x^{\prime })}){)}_{x^{\prime }}=0$, where $x^{\prime }$ is the projection of $x$ in $X^{\prime }$; if $U^{\prime }$ is the set of $x^{\prime }\in X^{\prime }$ such that $({\mathcal{H}}_1(({\mathcal{L}}_{\bullet }^{\prime }{)}_{f^{\prime }(x^{\prime })}){)}_{x^{\prime }}=0$, then $U$ is the inverse image of $U^{\prime }$ by the projection $X\to X^{\prime }$, which reduces, for the first assertion, to the case where $S$ is Noetherian. For the second assertion, one further remarks that $A$ is an inductive limit of its sub-rings $A_{\lambda }$ that are $A^{\prime }$-algebras of finite type; set $X_{\lambda }=X^{\prime }{\times }_{S^{\prime }}{S}_{\lambda }$, where $S_{\lambda }=\mathrm{Spec}(A_{\lambda })$, ${\mathcal{L}}_{\bullet }^{(\lambda )}={\mathcal{L}}_{\bullet }^{\prime }{\otimes }_{S^{\prime }}{S}_{\lambda }$, and let $u_{\lambda }=u^{\prime }\otimes 1:{\mathcal{L}}_2^{(\lambda )}\to {\mathcal{L}}_1^{(\lambda )}$, $v_{\lambda }=v^{\prime }\otimes 1:{\mathcal{L}}_1^{(\lambda )}\to {\mathcal{L}}_0^{(\lambda )}$, so that one has ${\mathcal{H}}_1({\mathcal{L}}_{\bullet }^{(\lambda )})=Ker(v_{\lambda })/Im(u_{\lambda })$. Now, since the functor lim is exact in the category of commutative groups, one has Ker(v) = lim Ker(v_λ), $Im(u)=\lim Im(u_{\lambda })$, and ${\mathcal{H}}_1({\mathcal{L}}_{\bullet })=\lim {\mathcal{H}}_1({\mathcal{L}}_{\bullet }^{(\lambda )})$. If one has supposed that ${\mathcal{L}}_1$ is $f$-flat and (by reducing to the case where $X=U$) that one has proved ${\mathcal{H}}_1({\mathcal{L}}_{\bullet }^{(\lambda )})=0$ for every $\lambda$, one will indeed deduce the assertion.

I) Suppose henceforth $S$ Noetherian. One knows (without flatness hypothesis on ${\mathcal{L}}_0$) that the set $U$ is constructible in $X$ (9.9.6). Using now $(0_{III},\mathrm{9.2.5})$, it remains to show that for every generization $x^{\prime }$ of $x\in U$ in $X$, one has also $x^{\prime }\in U$. The method exposed in (12.0.2) applies without change (taking into account the fact that for a prescheme $Z$ over a field $k$ and an extension $k^{\prime }$ of $k$, the projection $Z{\otimes }_k{k}^{\prime }\to Z$ is faithfully flat). One may therefore suppose that $S$ is the spectrum of a discrete valuation ring $A$, of which one denotes by $t$ a uniformizer, with $x$ above the closed point of $S$ and $x^{\prime }$ above the generic point of $S$. The hypothesis that ${\mathcal{L}}_0$ is $f$-flat entails that $t$ is ${\mathcal{L}}_0$-regular $(0_I,\mathrm{6.3.4})$. One is then reduced to proving the following lemma (where $B$, $M$, $N$, $P$ will be replaced by ${\mathcal{O}}_x$, $({\mathcal{L}}_2{)}_x$, $({\mathcal{L}}_1{)}_x$, and $({\mathcal{L}}_0{)}_x$ respectively):