Ch.4 §19. Regular immersions - Éléments de Géométrie Algébrique

§19. Regular and transversally regular immersions

The present section is devoted, on the one hand, to a study of regular immersions (16.9.2) $Y \to X$ of preschemes, notably in the case where $X$ and $Y$ are flat over the same prescheme $S$ ; on the other hand, it gives complements on $M$ -regular sequences and normal flatness, generalizing in particular flatness results established by H. Hironaka in his theory of resolution of singularities [35].

19.1. Properties of regular immersions

Proposition (19.1.1).

Let $X$ be a locally Noetherian prescheme, $j : Y \to X$ an immersion, $y$ a regular point of $Y$ . For $j$ to be a regular immersion at the point $y$ , it is necessary and sufficient that $X$ be regular at the point $j (y)$ .

Taking (16.9.10) into account, one is reduced to proving the

Corollary (19.1.2).

Let $A$ be a Noetherian local ring, $J$ an ideal of $A$ ; suppose that the quotient ring $B = A / J$ is regular. For $A$ to be regular, it is necessary and sufficient that the ideal $J$ be regular.

Indeed, if $A$ is regular, one knows (0, 17.1.9) that $J$ is generated by a part of a regular system of parameters of $A$ , hence $J$ is regular. Conversely, if $J$ is regular, and if $(f_{i})_{1 ⩽ i ⩽ n}$ is an $A$ -regular sequence generating $J$ , the $(f_{i})$ form a part of a system of parameters of $A$ (0, 16.4.1), so one deduces from (0, 17.1.7) that $A$ is regular.

Definition (19.1.3).

Let $X$ be a prescheme, $Y$ a sub-prescheme of $X$ , $U$ an open set of $X$ containing $Y$ and such that $Y$ is defined by an Ideal $I$ of $O_{U}$ ; suppose that $I / I^{2}$ is a locally free $(O_{U} / I)$ -Module (which is the case if $Y$ is quasi-regularly immersed in $X$ ). For every $y \in Y$ , we call transversal codimension of $Y$ in $X$ at the point $y$ , and denote $co d i m_{y}^{t} (Y, X)$ , the rank of the free $(O_{Y, y})$ -module $(I / I^{2})_{y}$ . If $f : Z \to X$ is an immersion of image $Y$ , we likewise call transversal codimension of $f$ at the point $z \in Z$ the transversal codimension in $X$ at the point $f (z)$ of the sub-prescheme $Y$ .

Proposition (19.1.4).

Let $X$ be a locally Noetherian prescheme, $Y$ a sub-prescheme of $X$ regularly immersed; then for every $y \in Y$ , one has

  (19.1.4.1)    codim_y^t(Y, X) = codim_y(Y, X).

By virtue of (5.1.3.2), $co d i m_{y} (Y, X)$ is equal to the smallest of the dimensions of the local rings $A = O_{X, z}$ , where $z$ is the generic point of an irreducible component of $Y$ containing

$y$ ; as such a point $z$ is contained in every neighbourhood of $y$ in $X$ , $I_{z} / I_{z}^{2}$ is a free $A$ -module of rank $n = co d i m_{y}^{t} (Y, X)$ ; it suffices to see that $dim (A) = n$ . Now, since $z$ is a maximal point of the sub-prescheme $Y$ defined by $I$ , one has $dim (O_{Y, z}) = 0$ , so $I_{z}$ is an ideal of definition of the Noetherian local ring $A$ ; furthermore, the hypothesis that $I$ is quasi-regular entails that $I_{z}^{k} / I_{z}^{k + 1}$ is a free $(A / I_{z})$ -module of rank $bin o m (n + k - 1, n - 1)$ ; if $r$ is the length of the Artinian ring $A / I_{z}$ , the length of $I_{z}^{k} / I_{z}^{k + 1}$ is therefore $r \cdot bin o m (n + k - 1, n - 1)$ ; the Hilbert-Samuel polynomial of $A$ for the $I_{z}$ -preadic filtration is therefore of degree $n$ , which proves the proposition (0, 16.2.3).

By virtue of (19.1.4), we shall henceforth say "codimension" instead of "transversal codimension" when dealing with a sub-prescheme $Y$ regularly immersed in $X$ (even when $X$ is not locally Noetherian).

Proposition (19.1.5).

(i) For an immersion $j : Y \to X$ to be open, it is necessary and sufficient that it be regular and of codimension 0 at every point.

(ii) Let $f : Y \to X$ be a regular (resp. quasi-regular) immersion, $g : X^{'} \to X$ a flat morphism, $Y^{'} = Y \times_{X} X^{'}$ ; then the morphism $f^{'} = f_{(X^{'})} : Y^{'} \to X^{'}$ is a regular (resp. quasi-regular) immersion, and for every $y^{'} \in Y^{'}$ , if $y$ is the projection of $y^{'}$ in $Y$ , the codimension of $f^{'}$ at the point $y^{'}$ is equal to the codimension of $f$ at the point $y$ . Conversely, if $f$ is an immersion and $g : X^{'} \to X$ a faithfully flat and quasi-compact morphism, then, if $f^{'}$ is a quasi-regular immersion, so is $f$ .

(iii) If $f : Y \to X$ and $g : Z \to Y$ are regular immersions, then $f \circ g : Z \to X$ is a regular immersion, and for every $z \in Z$ , the codimension of $f \circ g$ at the point $z$ is the sum of the codimension of $g$ at the point $z$ and of the codimension of $f$ at the point $g (z)$ . Moreover, the sequence of canonical homomorphisms (16.2.7.1)

$(19.1.5.1) 0 \to g * (N_{Y / X}) \to N_{Z / X} \to N_{Z / Y} \to 0$

is exact, and for every $z \in Z$ , there exists a neighbourhood of $z$ in $Z$ in which the restrictions of the homomorphisms of this sequence form a split sequence.

(iv) Let $X$ be a locally Noetherian prescheme, $f : Y \to X$ and $g : Z \to Y$ two immersions, $z$ a point of $Z$ . The following conditions are equivalent:

a) $g$ is a regular immersion at the point $z$ , and $f$ is a regular immersion at the point $g (z)$ .

b) $g$ and $f \circ g$ are regular immersions at the point $z$ .

c) $f \circ g$ is a regular immersion at the point $z$ , $f$ is a regular immersion at the point $g (z)$ , and the sequence of canonical homomorphisms (19.1.5.1), restricted to a sufficiently small open neighbourhood of $z$ in $Z$ , is exact and split.

(v) Let $X$ be a locally Noetherian prescheme, $f : Y \to X$ , $g : Z \to Y$ two immersions, $z$ a point of $Z$ ; suppose that $y = g (z)$ is a regular point of $Y$ and $x = f (g (z))$ a regular point of $X$ (which entails, by (19.1.1), that $f$ is a regular immersion at the point $g (z)$ ); then,

for $f \circ g$ to be a regular immersion at the point $z$ , it is necessary and sufficient that $g$ be a regular immersion at the point $z$ .

Assertion (i) is another way of expressing (16.1.10). To prove the first assertion of (ii), one may restrict to the case where $Y$ is a closed sub-prescheme of $X$ defined by a regular (resp. quasi-regular) Ideal $I$ ; then $Y^{'}$ is the closed sub-prescheme of $X^{'}$ defined by the Ideal $g * (I) \cdot O_{X^{'}}$ (I, 4.4.5), which is here identified with $g * (I)$ since $g$ is flat; by replacing $X$ by a sufficiently small affine open set, one may suppose that $I$ admits a regular (resp. quasi-regular) sequence of generators $f_{i}$ (sections of $O_{X}$ over $X$ ). If the sequence $(f_{i})$ is regular, so is the sequence $(f_{i} \otimes 1)$ (which generates $g * (I)$ ) by virtue of (0, 15.2.5); the analogous result for quasi-regular sequences follows at once from the flatness of $g$ and the criterion (16.9.4). Likewise, if $g$ is faithfully flat and quasi-compact, and if $g * (I)$ is quasi-regular, $I$ is of finite type by virtue of (2.5.2); by flatness, one has $g * (I) / g * (I)^{2} = g * (I / I^{2})$ , hence, since $g * (I / I^{2})$ is locally free by hypothesis (16.9.4), so is $I / I^{2}$ (2.5.2). Finally, condition (iii) of (16.9.4) relative to $g * (I)$ entails the same condition for $I$ by virtue of the faithful flatness of $g$ (2.2.7). One therefore concludes by (16.9.4) that $I$ is quasi-regular.

To prove (iii), one may likewise suppose that $Y$ and $Z$ are closed sub-preschemes of $X$ , defined respectively by Ideals $I$ , $J$ of $O_{X}$ with $I \subset J$ ; furthermore, one may suppose that there exist sections $f_{i}$ $(1 ⩽ i ⩽ m)$ of $I$ over $X$ generating $I$ and such that, for $1 ⩽ i ⩽ m$ , the endomorphism of $O_{X} / (\sum_{j < i} f_{j} O_{X})$ defined by $f_{i}$ is injective, and sections $f_{i}$ $(m + 1 ⩽ i ⩽ n)$ of $J / I$ over $X$ generating $J / I$ and such that, for $m + 1 ⩽ i ⩽ n$ , the endomorphism of $(O_{X} / I) / (\sum_{m < j < i} f_{j} (O_{X} / I))$ defined by $f_{i}$ is injective. For every $z \in Z$ , let $U$ be an open neighbourhood of $z$ in $X$ such that there exists a section $f_{i}$ of $J$ over $U$ whose class in $Γ (U, J / I)$ is $f_{i} ∣ U$ for $m + 1 ⩽ i ⩽ n$ ; then $(O_{X} / J) ∣ U = (O_{U} / (J ∣ U))$ is isomorphic to $O_{U} / ((I ∣ U) + \sum_{i = m + 1}^{n} f_{i} O_{U}) = O_{U} / (\sum_{i = 1}^{n} f_{i} O_{U})$ , and one therefore sees that the sequence $(f_{i})_{1 ⩽ i ⩽ n}$ is regular in $O_{U}$ ; as it generates $J ∣ U$ , this proves the first assertion of (iii). To prove the last assertion of (iii), one may restrict (with the same notations) to the case where the images $\overset{ˉ}{f}_{i}$ of the $f_{i}$ in $J / J^{2}$ (for $1 ⩽ i ⩽ n$ ) form a basis of this $(O_{X} / J)$ -Module (16.9.5); for the same reason, one may suppose that the $\overset{ˉ}{f}_{i}$ for $1 ⩽ i ⩽ m$ are the canonical images of elements of a basis of the $(O_{X} / J)$ -Module $g * (I / I^{2})$ , and that the canonical images $\tilde{f}_{i}$ of the $\overset{ˉ}{f}_{i}$ in $(J / I) / (J / I)^{2}$ for $m + 1 ⩽ i ⩽ n$ form a basis of this $(O_{X} / J)$ -Module. This completes the proof of (iii).

Let us pass to the proof of (iv). The fact that a) implies c) follows from (iii). Let us prove that c) implies a). Set again $A = O_{X, g (z)}$ , so that $(O_{Y, g (z)}) = A / J$ ,

$(O_{Z, z}) = A / K$ with $J \subset K$ . By hypothesis, $J$ and $K$ are regular ideals of $A$ and one has a split exact sequence

$0 \to J / J^{2} \to K / K^{2} \to (K / J) / (K / J)^{2} \to 0$

(cf. (16.2.7)). This entails that there exist elements $f_{j}$ $(1 ⩽ j ⩽ r + s)$ of $K$ whose images $\overset{ˉ}{f}_{j}$ in $K / K^{2}$ form a basis of this $(A / K)$ -module, and such that moreover the $f_{j}$ of index $j ⩽ r$ belong to $J$ and are such that their images in $J / J^{2}$ form a basis of this $(A / J)$ -module. The $f_{j}$ for $1 ⩽ j ⩽ r + s$ therefore generate $K$ , and the $f_{j}$ for $1 ⩽ j ⩽ r$ generate $J$ since $A$ is Noetherian (Bourbaki, Alg. comm., chap. II, §3, n° 2, prop. 5); as $K$ is a regular ideal, it follows from (16.9.5) that the sequence $(f_{j})_{1 ⩽ j ⩽ r + s}$ is regular. By definition, the images of $f_{r + 1}, \dots, f_{r + s}$ in $K / J$ therefore form a regular sequence in this $(A / J)$ -module, which completes the proof that c) entails a) in (iv).

Finally, taking (16.9.10) into account, it is immediate that (v), as well as the equivalence of a) and b) in (iv), result from the

Corollary (19.1.6).

Let $A$ be a Noetherian local ring, $J$ , $K$ two ideals of $A$ contained in its maximal ideal $m$ and such that $J \subset K$ , $A^{'} = A / J$ , $K^{'} = K / J$ . Then:

(i) If $J$ and $K^{'}$ are regular ideals, so is $K$ .

(ii) If $K$ and $K^{'}$ are regular ideals, so is $J$ .

(iii) Suppose that the rings $A$ and $A^{'} = A / J$ are regular. For the ideal $K$ to be regular, it is necessary and sufficient that $K^{'}$ be regular.

Assertion (i) is a particular case of (19.1.5, (iii)). To prove (ii), consider the canonical surjective homomorphism $u : K / K^{2} \to K^{'} / K^{' 2} = K / (K^{2} + J)$ . As by hypothesis $K / K^{2}$ and $K^{'} / K^{' 2}$ are free $(A / K)$ -modules, whose respective ranks we denote $p + q$ and $q$ , the kernel of $u$ is a projective $(A / K)$ -module (Bourbaki, Alg., chap. II, 3rd ed., §2, n° 2, prop. 4), hence free of rank $p$ since $A / K$ is a Noetherian local ring $(0_{III}, 10.1.3)$ ; in other words, there exists a basis of $K / K^{2}$ whose $p$ first elements form a basis of $(K^{2} + J) / K^{2}$ . This means again (by virtue of Nakayama's lemma) that there exists a system of generators $(f_{i})_{1 ⩽ i ⩽ p + q}$ of $K$ , forming a regular sequence in $A$ , such that the sequence $(f_{i})_{1 ⩽ i ⩽ p}$ is formed of elements of $J$ ; we must show that this latter sequence generates $J$ . To do so, denote by $J^{'} \subset J$ the ideal it generates; by considering the ring $A / J^{'}$ , one sees that one is reduced to proving the following lemma:

Lemma (19.1.6.1).

Let $B$ be a Noetherian local ring of maximal ideal $n$ , $(g_{j})_{1 ⩽ j ⩽ q}$ a regular sequence in $B$ of elements of $n$ , $b$ the ideal it generates, $c \subset b$ an ideal such that the canonical images of the $g_{j}$ in $B / c$ again form a regular sequence in this ring. Then one has $c = 0$ .

The lemma being trivial for $q = 0$ , let us reason by induction on $q$ . The induction hypothesis, applied to the ring $B / g_{1} B$ and to the canonical images of the $g_{j}$ $(2 ⩽ j ⩽ q)$

and of $c$ in this quotient ring, shows that $c \subset g_{1} B$ . Let $b^{'}$ be the ideal of $B$ formed by $x \in B$ such that $g_{1} x \in c$ ; one then has $c = g_{1} b^{'}$ ; but since by hypothesis $g_{1}$ is regular in $B / c$ , one necessarily has $b^{'} = c$ , hence $c = g_{1} c$ . As $c$ is of finite type and $g_{1}$ is contained in the radical of $B$ , Nakayama's lemma shows that $c = 0$ .

Let us finally prove assertion (iii) of (19.1.6). By virtue of (i) and of (19.1.2), it suffices to see that if $K^{'}$ is regular, so is $K$ . Note that $J$ is regular (19.1.2), and let us reason by induction on the rank $q$ of the free $(A / J)$ -module $J / J^{2}$ . If $q = 0$ , one has $J = 0$ by Nakayama's lemma and the assertion is trivial. As $A$ and $A / J$ are regular, one knows (0, 17.1.9) that $J$ is generated by a part with $q$ elements of a regular system of parameters of $A$ ; so, if $f$ is one of the elements of this system of generators of $J$ , $A_{1} = A / f A$ is regular and $f \in / m^{2}$ (0, 17.1.8). Let $J_{1}$ , $K_{1}$ be the canonical images of $J$ , $K$ in A_1; one has $J_{1} \subset K_{1}$ , $A_{1} / J_{1} = A / J$ , so $A_{1} / J_{1}$ is regular and consequently $J_{1}$ is a regular ideal (19.1.2). Let us show that $K$ is a regular ideal in A_1; as $K$ is a regular ideal in $A$ , it suffices to show that $f$ is part of a regular system of generators of $K$ , and for this (16.9.5) it suffices to show that the image of $f$ in $K / (K^{2} + mK) = K / mK$ is part of a basis of this $(A / m)$ -vector space, in other words that $f \in / mK$ , which indeed follows from $f \in / m^{2}$ . Then, as $J_{1} / J_{1}^{2}$ is a free $(A_{1} / J_{1})$ -module of rank $q - 1$ , the induction hypothesis shows that $K_{1}^{'} = K_{1} / J_{1}$ is a regular ideal of $A_{1} / J_{1}$ , and as $K^{'}$ is identified with $K_{1}^{'}$ in the canonical isomorphism between $A / J$ and $A_{1} / J_{1}$ , $K^{'}$ is regular. Q.E.D.

Remarks (19.1.7).

(i) We do not know whether the composite of two quasi-regular immersions is quasi-regular.

(ii) In a ringed space in local rings $X$ , for every Ideal $I$ of $O_{X}$ , $O_{X} / I$ is of finite type and so has closed support in $X$ $(0_{I}, 5.2.2)$ . One may consequently define, as in (I, 4.1.3 and 4.2.1), the notions of (locally closed) ringed sub-space of $X$ defined by an Ideal $I$ of a sheaf $O_{U}$ , where $U$ is an open set of $X$ , and the notion of immersion. The definitions of quasi-regular and regular immersions, of transversal codimension then apply without modification, and the results of (19.1.5, (i) to (iv)) are still valid, supposing in the second assertion of (i) and in (iv) that the sheaf $O_{X}$ is coherent and the local rings $O_{X, x}$ Noetherian; as for (ii), one must define $Y^{'}$ as the ringed sub-space defined by the $O_{X^{'}}$ -Ideal $g * (I)$ ; the proofs are unchanged.

(19.1.8)

We have already pointed out (notably in (16.9.6, (ii))) that in non-Noetherian rings, the notion of "regular sequence" of elements of the ring does not possess the properties that would make it usable. Recent research seems to show that the notion that should in this case replace that of regular sequence is that of a sequence $f = (f_{1}, \dots, f_{n})$ having the property that the exterior-algebra complex $K_{∙} (f, M)$ (III, 1.1.3) has its homology zero in degrees > 0 (cf. (III, 1.1.4)); see in particular A. Grothendieck, Séminaire de géométrie algébrique, 1966, to appear shortly.

19.2. Transversally regular immersions

Definition (19.2.1).

Let $u : X \to S$ be a morphism of preschemes, $F$ a quasi-coherent $O_{X}$ -Module. We say that a sequence $(f_{i})_{1 ⩽ i ⩽ n}$ of sections of $O_{X}$ over $X$ is transversally $F$ -regular relatively to $S$ (or relatively to $u$ ) if the sequence $(f_{i})$ is $F$ -regular and such that the $O_{X}$ -Modules $F / (\sum_{j ⩽ i} f_{j} F)$ are $S$ -flat for $0 ⩽ i ⩽ n$ . We say that a quasi-coherent Ideal $I$ of $O_{X}$ is transversally regular relatively to $S$ (or relatively to $u$ ) at a point $x \in S u pp (O_{X} / I)$ if there exists an open neighbourhood $U$ of $x$ and a finite sequence $(f_{i})$ of sections of $I$ over $U$ which is transversally $O_{U}$ -regular relatively to $S$ and which generates $I ∣ U$ .

In an $A$ -algebra $B$ , we say that an ideal $J$ is transversally regular relatively to $A$ if $I = \tilde{J}$ is transversally regular relatively to $S = Spec (A)$ at every point of $V (J)$ in $Spec (B)$ .

Definition (19.2.2).

Let $S$ be a prescheme, $X$ , $Y$ two $S$ -preschemes, $u : Y \to X$ an $S$ -morphism which is an immersion. We say that $u$ is a transversally regular immersion relatively to $S$ at a point $y \in Y$ if there exists a neighbourhood $V$ of $u (y)$ in $X$ such that the sub-prescheme induced on $u (Y) \cap V$ by the sub-prescheme of $X$ associated to $u$ is defined by an Ideal of $O_{V}$ transversally regular relatively to $S$ at the point $u (y)$ .

It is clear that the set of points of $Y$ where an immersion is transversally regular relatively to $S$ is open in $Y$ ; if it equals $Y$ , one says simply that $u$ is a transversally regular immersion relatively to $S$ .

If $S \to T$ is a flat morphism, then if $I$ is an Ideal of $O_{X}$ transversally regular relatively to $S$ at $x \in X$ , it is so also relatively to $T$ $(0_{I}, 6.2.1)$ . An $S$ -immersion $Y \to X$ transversally regular at a point $y \in Y$ relatively to $S$ is therefore also a $T$ -immersion transversally regular relatively to $T$ at this point.

Proposition (19.2.3).

Let $S$ be a prescheme, $X$ , $Y$ two $S$ -preschemes, $u : Y \to X$ an $S$ -morphism which is a transversally regular immersion relatively to $S$ at a point $y \in Y$ . Then, for every base change $S^{'} \to S$ , if one sets $X^{'} = X \times_{S} S^{'}$ , $Y^{'} = Y \times_{S} S^{'}$ , the immersion $u^{'} = u_{(S^{'})} : Y^{'} \to X^{'}$ is transversally regular relatively to $S^{'}$ at every point $y^{'} \in Y^{'}$ above $y$ .

This follows at once from the definition and from (0, 15.1.15).

Proposition (19.2.4).

Let $S$ be a prescheme, $X$ , $Y$ two $S$ -preschemes, $u : Y \to X$ an $S$ -morphism which is an immersion. Suppose either that $S$ and $X$ are locally Noetherian, or that the structure morphisms $g : X \to S$ , $h : Y \to S$ are locally of finite presentation. Let $y$ be a point of $Y$ , $s$ its image in $S$ . The following conditions are equivalent:

a) $u$ is transversally regular relatively to $S$ at the point $y$ .

b) The morphisms $g$ and $h$ are flat in a neighbourhood of the points $u (y)$ and $y$ respectively, and if one sets $X_{s} = g^{- 1} (s)$ , $Y_{s} = h^{- 1} (s)$ , the immersion $u_{s} : Y_{s} \to X_{s}$ is regular at the point $y$ .

b') The morphisms $g$ and $h$ are flat in a neighbourhood of the points $u (y)$ and $y$ respectively, and, for every base change $S^{'} \to S$ , if one sets $X^{'} = X \times_{S} S^{'}$ , $Y^{'} = Y \times_{S} S^{'}$ , the immersion $u^{'} = u_{(S^{'})} : Y^{'} \to X^{'}$ is regular at every point of $Y^{'}$ above $y$ .

c) The morphism $h$ is flat in a neighbourhood of the point $y$ , and the immersion $u$ is regular in a neighbourhood of the point $y$ .

c') The morphism $h$ is flat in a neighbourhood of the point $y$ , and the immersion $u$ is quasi-regular in a neighbourhood of the point $y$ .

It was already proved in (19.2.3) that a) entails b') without finiteness hypothesis, and it is trivial that b') entails b). Let us show that b) entails c). For this (the question being local on $X$ and on $S$ ) one may suppose that $Y$ is a closed sub-prescheme of $X$ defined by a quasi-coherent Ideal $I$ of $O_{X}$ , and $u$ the canonical injection. The hypothesis that $O_{X} / I$ is $S$ -flat entails that the sequence

  0 → 𝓘 ⊗_{𝒪_S} k(s) → 𝒪_X ⊗_{𝒪_S} k(s) → (𝒪_X/𝓘) ⊗_{𝒪_S} k(s) → 0

is exact $(0_{I}, 6.1.2)$ , so $Y_{s}$ is the closed sub-prescheme of $X_{s}$ defined by the Ideal $I_{s}$ of $O_{X_{s}} = O_{X} \otimes_{O_{S}} k (s)$ which is identified with $I \otimes_{O_{S}} k (s) = I / m_{s} I$ . Consider a finite sequence $(f_{i})$ of sections of $I$ over a neighbourhood of $y$ in $X$ whose images in $I_{y} / (m_{s} O_{X, y} \cdot I_{y} + I_{y}^{2})$ form a basis of this $(O_{X, y} / (m_{s} O_{X, y} + I_{y}))$ -module (which is free by virtue of the hypothesis b)). Since $X_{s}$ is locally Noetherian, it follows from (16.9.11), (16.9.5) and the hypothesis b) that the images $(\overset{ˉ}{f}_{i})_{y} = (f_{i})_{y} \otimes 1$ , sections of $I_{s} = I \otimes_{O_{S}} k (s)$ over a neighbourhood of $y$ in $X_{s}$ , form a regular sequence generating the restriction of $I_{s}$ to this neighbourhood. As $m_{s} O_{X, y}$ is contained in the radical of $O_{X, y}$ , and as in both cases considered $I_{y}$ is an ideal of finite type of $O_{X, y}$ , it follows from Nakayama's lemma that the $(f_{i})_{y}$ also generate $I_{y}$ ; as $I$ is an Ideal of finite type of $O_{X}$ in both cases considered, there is a neighbourhood $U$ of $y$ in $X$ such that the $f_{i} ∣ U$ generate $I ∣ U$ $(0_{I}, 5.2.2)$ . The fact that b) entails c) is then a consequence of (0, 15.1.16) when $S$ and $X$ are locally Noetherian, and of (11.3.8) when $g$ and $h$ are locally of finite presentation; in this latter case, (11.3.8) also proves the equivalence of c) and c') (which is trivial when $X$ and $S$ are locally Noetherian (0, 15.1.11)). Finally, if c) is verified, to prove a) one may again restrict to the case where $Y$ is a closed sub-prescheme of $X$ defined by $I$ , and by hypothesis there is a regular sequence $(f_{i})$ of sections of $I$ over a neighbourhood $U$ of $y$ in $X$ such that the $f_{i}$ generate $I ∣ U$ and $h$ is flat in $U$ ; to see that c) entails a), it again suffices to apply (0, 15.1.6) when $S$ and $X$ are locally Noetherian, and (11.3.8) when $g$ and $h$ are locally of finite presentation.

Remarks (19.2.5).

(i) When in condition b), one suppresses the hypothesis that $h$ is flat in a neighbourhood of $y$ , the thus modified condition no longer entails a). It suffices to take for $S$ the spectrum of a local ring which is not a field, for $s$ the closed point of $S$ , and $Y = X_{s}$ .

(ii) Suppose that $Y$ is a closed sub-prescheme of $X$ defined by a quasi-coherent Ideal $I$ of $O_{X}$ . It was proved in the course of the proof of (19.2.4) that, when the equivalent conditions of this proposition are fulfilled, then, for every system $(f_{i})$ of sections of $I$ over a neighbourhood of $y$ in $X$ such that the images of the $f_{i}$ in $I_{y} / (m_{s} O_{X, y} \cdot I_{y} + I_{y}^{2})$ form a basis of this $(O_{X, y} / (m_{s} O_{X, y} + I_{y}))$ -module, there exists an open neighbourhood $U$ of $y$ in $X$ such that the $f_{i} ∣ U$ satisfy the conditions of definition (19.2.1).

Corollary (19.2.6).

Suppose either that $S$ and $X$ are locally Noetherian, or that $g$ and $h$ are locally of finite presentation. Suppose moreover that the fibres $X_{s}$ and $Y_{s}$ are regular preschemes at the points $u (y)$ and $y$ respectively, and that the morphisms $g$ and $h$ are flat in a neighbourhood of $u (y)$ and $y$ respectively. Then the immersion $Y \to X$ is transversally regular at the point $y$ .

Indeed, it follows from (19.1.1) that the immersion $Y_{s} \to X_{s}$ is regular at the point $y$ , and it suffices to apply (19.2.4, b)).

Proposition (19.2.7).

(i) Every $S$ -open immersion is transversally regular relatively to $S$ .

(ii) Let $f : Y \to X$ be an $S$ -immersion, $g : S^{'} \to S$ a morphism, and set $X^{'} = X_{(S^{'})}$ , $Y^{'} = Y_{(S^{'})}$ , $f^{'} = f_{(S^{'})} : Y^{'} \to X^{'}$ . If $f$ is a transversally regular immersion relatively to $S$ , $f^{'}$ is a transversally regular immersion relatively to $S^{'}$ . The converse is true if $g$ is faithfully flat, and moreover if $X$ and $Y$ are locally of finite presentation over $S$ .

(iii) If $f : Y \to X$ and $g : Z \to Y$ are transversally regular immersions relatively to $S$ , so is $f \circ g : Z \to X$ .

(iv) Let $X$ be an $S$ -prescheme, $f : Y \to X$ , $g : Z \to Y$ two $S$ -immersions. Suppose either that $S$ and $X$ are locally Noetherian, or that $X$ , $Y$ and $Z$ are locally of finite presentation over $S$ . Then, if $g$ and $f \circ g$ are transversally regular at the point $z \in Z$ relatively to $S$ , $f$ is transversally regular at the point $g (z)$ relatively to $S$ .

(v) Under the general conditions of (iv), suppose that $X$ and $Y$ are $S$ -flat, $Y_{s}$ regular at the point $g (z)$ and $X_{s}$ regular at the point $g (z)$ ; then, if $f \circ g$ is transversally regular at the point $z$ , so is $g$ .

Assertion (i) is immediate, and the first assertion of (ii) follows at once from (19.2.3). To prove the second assertion of (ii), let us apply (19.2.4): $X^{'}$ and $Y^{'}$ being locally of finite presentation over $S^{'}$ , it follows first from this criterion that $X^{'}$ and $Y^{'}$ are flat over $S^{'}$ , so $X$ and $Y$ are flat over $S$ (2.5.1); on the other hand, for every $s \in S$ , if $s^{'} \in S^{'}$ is a point above $s$ , $Y_{s^{'}}^{'} \to X_{s^{'}}^{'}$ is by hypothesis a regular immersion, and one deduces from (19.1.5, (ii)) that the same holds for $Y_{s} \to X_{s}$ , since these preschemes are locally Noetherian and it then amounts to the same to say that the immersion $Y_{s} \to X_{s}$ is regular or quasi-regular. To prove (iii), one reduces to the case where $Z$ is a closed sub-prescheme of $Y$ and $Y$ a closed sub-prescheme of $X$ , and one forms as in (19.1.5, (iii)) a system of generators of the Ideal of $O_{X}$ defining $Z$ . To prove (iv), one notes that the hypothesis combined with (19.2.4) shows first that $X$ , $Y$ and $Z$ are $S$ -flat in a neighbourhood of $z$ , and it suffices moreover, if $s$ is the image of $z$ in $S$ , to prove that the immersion $f_{s} : Y_{s} \to X_{s}$ is regular at the point $z$ , knowing that the same holds for $g_{s} : Z_{s} \to Y_{s}$ and for $f_{s} \circ g_{s}$ ; but this follows from (19.1.5, (iv)). Finally, for (v), one reasons in the same way as for (iv); the hypothesis on $f \circ g$ already implies by (19.2.4, b)) that $Z$ is $S$ -flat in a neighbourhood of $z$ , and the same holds for $X$ and $Y$ by hypothesis; one is then reduced to proving that $g_{s}$ is regular at the point $z$ , knowing that the same holds for $f_{s} \circ g_{s}$ , which follows from (19.1.5, (v)), given the hypotheses made on $X_{s}$ and $Y_{s}$ .

Proposition (19.2.8).

Let $X$ be an $S$ -prescheme, $Y$ a closed sub-prescheme of $X$ defined by a quasi-coherent Ideal $I$ of $O_{X}$ , $u : Y \to X$ the canonical injection. If $u$ is transversally regular relatively to $S$ , the $O_{X}$ -Modules $I^{n}$ and $I^{n} / I^{m}$ $(0 ⩽ n < m)$ are $S$ -flat at the points of $Y$ . For every base change $S^{'} \to S$ , if $X^{'} = X_{(S^{'})}$ , $Y^{'} = Y_{(S^{'})}$ and $u^{'} = u_{(S^{'})}$ , one has $I_{(u^{'})}^{n} = I_{(u)}^{n} \otimes_{O_{S}} O_{S^{'}}$ up to isomorphism, for every $n > 0$ .

Indeed, $O_{X}$ and $O_{X} / I$ are by hypothesis $S$ -flat at the points of $Y$ , and $I^{n} / I^{n + 1}$ is a locally free $(O_{X} / I)$ -module (16.9.3), so it is $S$ -flat at the points of $Y$ . The first assertion then follows from $(0_{I}, 6.1.2)$ and the second from (11.2.9.2).

The following proposition generalizes and refines (17.16.1):

Proposition (19.2.9).

Let $f : X \to S$ be a morphism, $x$ a point of $X$ , $s = f (x)$ . Suppose that $f$ is of finite presentation at the point $x$ , and is a Cohen-Macaulay morphism at the point $x$ (6.8.1). Then there exists a sub-prescheme $Y$ of $X$ containing $x$ and having the following properties:

(i) The canonical injection $j : Y \to X$ is transversally regular relatively to $S$ .

(ii) The morphism $g = f \circ j : Y \to S$ is Cohen-Macaulay (which amounts to saying, taking (i) into account, that all the fibres $g^{- 1} (f (x))$ of $g$ are Cohen-Macaulay preschemes).

(iii) The point $x$ is a maximal point of $Y_{s} = g^{- 1} (s)$ .

Set $X_{s} = f^{- 1} (s)$ ; by hypothesis, the ring $O_{X_{s}, x}$ is Cohen-Macaulay; let $(t_{i})_{1 ⩽ i ⩽ n}$ be a system of parameters of this ring, which is therefore a regular sequence (0, 16.5.7). There is an open neighbourhood $U$ of $x$ in $X$ and sections $h_{i}$ $(1 ⩽ i ⩽ n)$ of $O_{X}$ over $U$ such that the $t_{i}$ are the canonical images of the $(h_{i})_{x}$ in the quotient ring $O_{X_{s}, x}$ of $O_{X, x}$ . Let $I$ be the Ideal of $O_{U}$ generated by the $h_{i}$ and take for $Y$ the closed sub-prescheme of $U$ defined by $I$ . One may suppose that $f ∣ U$ is locally of finite presentation, and it is therefore the same for $g$ by definition of $Y$ ; it follows then from the equivalence of c') and a) in (11.3.8) that the sequence $((h_{i})_{x})$ is regular in $O_{X, x}$ and that $O_{Y, x}$ is flat over $O_{S, s}$ ; by virtue of (11.3.1), $f$ and $g$ are therefore flat in a neighbourhood of $x$ and it follows from (19.2.4) that $j$ is a transversally regular immersion relatively to $S$ (replacing if necessary $Y$ by $Y \cap V$ for an open neighbourhood $V$ of $x$ in $X$ ). One has thus proved (i) and one sees that one may suppose that $g : Y \to S$ is a flat morphism. As $O_{Y_{s}, x}$ , quotient of $O_{X_{s}, x}$ by the ideal generated by the $t_{i}$ , is Artinian by construction (0, 16.3.6), $x$ is a maximal point of $Y_{s}$ . Finally, the ring $O_{Y_{s}, x}$ being Artinian is a Cohen-Macaulay ring (0, 16.5.1), so, by replacing if necessary $Y$ by its intersection with an open neighbourhood of $x$ in $X$ , one may suppose that $g$ is a Cohen-Macaulay morphism, by virtue of (12.1.7, (iii)).

In particular, one recovers the existence of "flat quasi-sections" $Y$ of a morphism $f : X \to S$ at a point $x \in X_{s}$ which is closed in $X_{s}$ and such that $f$ is flat at the point $x$ and $O_{X_{s}, x}$ is a Cohen-Macaulay ring (17.16.1), and one sees moreover that the immersion $Y \to X$ is transversally regular relatively to $S$ . This last complement is also valid for the étale quasi-section defined in (17.16.3, (i)).

19.3. Relative complete intersections (flat case)

Definition (19.3.1).

Let $A$ be a Noetherian local ring. We say that $A$ is a complete intersection ring (or also absolute complete intersection ring, when there is danger of confusion) if the completion Â is isomorphic to the quotient of a complete regular Noetherian local ring $B$ by a regular ideal (i.e. (16.9.7) an ideal generated by a regular sequence of elements of $B$ ).

Every regular local ring is a complete intersection ring.

One says that a locally Noetherian prescheme $X$ is a (absolute) complete intersection at the point $x \in X$ if the ring $O_{X, x}$ is a complete intersection ring.

Proposition (19.3.2).

Let $B$ be a regular Noetherian local ring, $J$ an ideal of $B$ . For $A = B / J$ to be a complete intersection ring, it is necessary and sufficient that the ideal $J$ be regular (i.e. (16.9.7) generated by a regular sequence of elements of $B$ ).

One may write $\hat{A} = \hat{B} / J \hat{B}$ and since $\hat{B}$ is a faithfully flat $B$ -module and $\hat{B}$ Noetherian, $J \hat{B}$ is regular if and only if $J$ is (19.1.5, (ii)). This shows (by virtue of (0, 17.1.5)) on the one hand that the condition of the statement is sufficient, and on the other hand that to prove it is necessary, one may restrict to the case where $A$ and $B$ are complete. By hypothesis, there then exists a complete regular Noetherian local ring $B^{'}$ such that $A$ is isomorphic to a quotient $B^{'} / J^{'}$ , where $J^{'}$ is a regular ideal of $B^{'}$ . Consider then the fibre product $B \times_{A} B^{'} = B^{''}$ for the surjective homomorphisms $B \to A$ , $B^{'} \to A$ (0, 18.1.2); we shall first prove the following lemma:

Lemma (19.3.2.1).

Let $A$ , $E$ , $F$ be three rings, $f : E \to A$ , $g : F \to A$ two homomorphisms, and set $G = E \times_{A} F$ (0, 18.1.2).

(i) If $f$ is surjective and if $E$ and $F$ are Noetherian rings, so is $G$ .

(ii) If $A$ , $E$ and $F$ are local rings and $f$ and $g$ are local homomorphisms, $G$ is a local ring and the canonical homomorphisms $G \to E$ , $G \to F$ are local.

(iii) If the conditions of (i) and (ii) are verified, if $g$ is surjective, and if $E$ and $F$ are complete Noetherian local rings, so is $G$ .

(i) Taking into account (0, 18.1.3 and 18.1.5), $G$ is an $A$ -ring augmented over $F$ , whose augmentation ideal $J^{'}$ is canonically identified with the kernel $J$ of $f$ , the $G$ -module structure on $J^{'}$ being identified with that of $E$ -module on $J$ via the canonical homomorphism $G \to E$ . If $a$ is any ideal of $G$ , $a / (a \cap J^{'})$ is isomorphic to $(a + J^{'}) / J^{'}$ , hence to an ideal of $F$ , and is therefore of finite type; on the other hand, $a \cap J^{'}$ is isomorphic to an ideal of $E$ contained in $J$ , hence is also of finite type, which proves that $a$ is of finite type, so $G$ is Noetherian.

(ii) Let $r$ , $s$ be the maximal ideals of $E$ and $F$ respectively; the set $m = r \times_{A} s$ is evidently an ideal of $G$ , the inverse image of $r$ by the projection $G \to E$ and of $s$ by the projection $G \to F$ . One deduces at once that the elements of $m$ are non-invertible in $G$ . On the other hand, if $(x, y) \in / m$ , and if for instance $x \in / r$ , $f (x)$ does not belong to the maximal ideal of $A$ , hence, since $g (y) = f (x)$ , and since $g$ is local, one has also $y \in / s$ ; one

concludes that $x$ and $y$ are invertible, so the same holds for $(x, y)$ , which completes the proof of (ii).

(iii) The image of $m$ by the surjective homomorphism $G \to E$ is the maximal ideal $r$ of $E$ , so the image of $m \cap J^{'}$ is $r \cap J$ , and as $J$ is closed (hence complete) for the topology induced by the $r$ -adic topology of $E$ $(0_{I}, 7.3.5)$ , $J^{'}$ is complete for the topology induced by the $m$ -preadic topology of $G$ . On the other hand, $G / J^{'}$ , endowed with the $m$ -preadic topology, is isomorphic to $F$ endowed with the $s$ -adic topology, hence is complete for the quotient topology of the $m$ -preadic topology of $G$ . One deduces at once that $G$ is complete for the $m$ -preadic topology.

This lemma being established, it follows from Cohen's theorem (0, 19.8.8) that B'' is isomorphic to a quotient of a complete regular Noetherian local ring $C$ . It then follows from (19.1.2) that the immersion $Spec (B^{'}) \to Spec (C)$ is regular; as by hypothesis the same holds for the immersion $Spec (A) \to Spec (B^{'})$ , one concludes from (19.1.5, (iii)) that the immersion $Spec (A) \to Spec (C)$ is regular; but this immersion is also written as the composite Spec(A) → Spec(B) → Spec(C). Now as $B$ is regular by hypothesis, the immersion $Spec (B) \to Spec (C)$ is regular (19.1.2); so the same holds for $Spec (A) \to Spec (B)$ by (19.1.5, (iv)).

Corollary (19.3.3).

Let $X$ be a locally Noetherian prescheme locally immersible in a regular prescheme. Then the set of $x \in X$ where $X$ is a complete intersection is open in $X$ and contains the set of regular points of $X$ .

One may indeed restrict to the case where $X = Spec (B / J)$ , where $B$ is a regular ring and $J$ an ideal of $B$ . The set of $p \supset J$ in $Spec (B)$ such that $J_{p}$ is regular in $B_{p}$ is then open (16.9.5), and it is the set of points of $Spec (B)$ where $X$ is a complete intersection by virtue of (19.3.2).

Corollary (19.3.4).

Let $k$ be a field, $X$ a prescheme locally of finite type over $k$ , $k^{'}$ an extension of $k$ , $X^{'} = X \otimes_{k} k^{'}$ . Let $x^{'}$ be a point of $X^{'}$ , $x$ its projection in $X$ . For $X$ to be a complete intersection at $x$ , it is necessary and sufficient that $X^{'}$ be a complete intersection at $x^{'}$ .

The question being local on $X$ , one may suppose that $X = Spec (A)$ , where $A$ is a $k$ -algebra of finite type, hence a quotient of a polynomial algebra $k [T_{1}, \dots, T_{n}]$ , so that $X$ is a closed sub-prescheme of the regular scheme $Y = Spec (k [T_{1}, \dots, T_{n}])$ (0, 17.3.7). If one sets $Y^{'} = Y \otimes_{k} k^{'} = Spec (k^{'} [T_{1}, \dots, T_{n}])$ , $Y^{'}$ is also a regular scheme. Now, since $Spec (k^{'})$ is faithfully flat over $Spec (k)$ , it follows from (19.1.5, (ii)) and from the fact that these are Noetherian preschemes that the immersion $X \to Y$ is regular at the point $x$ if and only if the immersion $X^{'} \to Y^{'}$ is regular at the point $x^{'}$ . The conclusion therefore follows from the criterion (19.3.2).

Corollary (19.3.5).

Let $A$ , $C$ be two Noetherian local rings, $ϕ : A \to C$ a local homomorphism, $B = C / J$ a quotient ring of $C$ , $k$ the residue field of $A$ . Suppose that $ϕ$ makes $C$ a flat $A$ -module, and that the ring $C \otimes_{A} k$ is regular. Then the following conditions are equivalent:

a) The ideal $J$ is transversally regular relatively to $A$ .

b) $B$ is a flat $A$ -module, and $B \otimes_{A} k$ is a complete intersection ring.

By virtue of (19.2.4), condition a) amounts to saying that $B$ is a flat $A$ -module and that $J \otimes_{A} k$ (which is identified with an ideal of $C \otimes_{A} k$ by virtue of the flatness of $B$ over $A$ $(0_{I}, 6.1.2)$ ) is regular in this ring. Since the ring $C \otimes_{A} k$ is regular, it amounts to the same, by virtue of (19.3.2), to say (when $B$ is a flat $A$ -module) that $J \otimes_{A} k$ is a regular ideal of $C \otimes_{A} k$ , or that $B \otimes_{A} k = (C \otimes_{A} k) / (J \otimes_{A} k)$ is a complete intersection ring; whence the corollary.

Definition (19.3.6).

Let $f : X \to S$ be a flat morphism locally of finite presentation. We say that $X$ is a relative complete intersection relatively to $S$ at a point $x$ if the fibre $f^{- 1} (f (x))$ is an (absolute) complete intersection at the point $x$ . We say that $X$ is a relative complete intersection relatively to $S$ , or that the morphism $f$ is a complete intersection morphism, if $X$ is a relative complete intersection relatively to $S$ at each of its points.

Proposition (19.3.7).

Let $g : X \to S$ , $h : Y \to S$ be two flat morphisms locally of finite presentation, $f : Y \to X$ an $S$ -immersion, $y$ a point of $Y$ , $x = f (y)$ , $s = g (x) = h (y)$ . Suppose that the fibre $X_{s} = g^{- 1} (s)$ is regular at the point $x$ . Then, for $f$ to be a transversally regular immersion at the point $y$ , it is necessary and sufficient that $Y$ be a relative complete intersection relatively to $S$ at the point $y$ .

To say that $f$ is transversally regular at the point $y$ amounts, by virtue of (19.2.4), to saying that $f_{s} : Y_{s} \to X_{s}$ is a regular immersion at the point $y$ . But as $X_{s}$ is regular at the point $x = f (y)$ , this is equivalent, by virtue of (19.3.2), to saying that $Y_{s}$ is an (absolute) complete intersection at the point $y$ , whence the proposition.

Corollary (19.3.8).

Let $f : X \to S$ be a flat morphism locally of finite presentation. The set $U$ of $x \in X$ at which $X$ is a relative complete intersection relatively to $S$ is open in $X$ . If moreover $f$ is proper, the set $E$ of $s \in S$ such that $X_{s} = f^{- 1} (s)$ is a complete intersection at each of its points is an open part of $S$ .

As $E = S - f (X - U)$ , the second assertion follows from the first and from the fact that $f$ is a closed map. The first assertion is local on $X$ , so one may suppose that $X$ is a closed sub-prescheme of a prescheme of the form $Z = S [T_{1}, \dots, T_{n}]$ (the $T_{i}$ indeterminates). This latter being smooth over $S$ (17.3.8), its fibres $Z_{s}$ are regular, and it amounts to the same, by virtue of (19.3.7), to say that $x \in U$ or that the immersion $X \to Z$ is transversally regular at the point $x$ ; one therefore concludes the corollary by (19.2.2).

Proposition (19.3.9).

(i) Every open immersion is a complete intersection morphism.

(ii) Let $f : X \to S$ be a flat morphism locally of finite presentation which is a complete intersection morphism. For every base change $g : S^{'} \to S$ , $f^{'} = f_{(S^{'})} : X_{(S^{'})} \to S^{'}$ is a complete intersection morphism. The converse is true if $g$ is faithfully flat and quasi-compact.

(iii) If $f : X \to Y$ and $g : Y \to Z$ are two flat morphisms locally of finite presentation which are complete intersection morphisms, so is $g \circ f : X \to Z$ .

Assertion (i) follows at once from definition (19.3.1). To prove (ii), let us note first that if $f$ is flat and locally of finite presentation, so is

$f^{'}$ (2.1.4), the converse being true if $g$ is faithfully flat ((2.5.1) and (2.7.1)); furthermore, for every $s^{'} \in S^{'}$ , if $s = g (s^{'})$ , it is equivalent to say that $f^{- 1} (s)$ is a complete intersection or that $f^{' - 1} (s^{'})$ is a complete intersection (19.3.4); whence (ii), taking into account that $g$ is surjective when it is faithfully flat.

Finally, under the hypotheses of (iii), $g \circ f$ is flat and locally of finite presentation. By definition (19.3.6), one is therefore reduced to the case where $Z$ is the spectrum of a field, and to proving then that the local rings $O_{X, x}$ are complete intersection rings, which is contained in the more general following result:

Corollary (19.3.10).

Let $f : X \to S$ be a flat morphism locally of finite presentation which is a complete intersection morphism, $x$ a point of $X$ , $s = f (x)$ . If $O_{S, s}$ is a Noetherian complete intersection ring, so is $O_{X, x}$ .

One may obviously restrict to the case where $S = Spec (A)$ with $A = O_{S, s}$ . Let us show moreover that one may restrict to the case where the local ring $A$ is complete. Indeed, set $A^{'} = \hat{A}$ , $X^{'} = X \otimes_{A} A^{'}$ , and let $s^{'}$ be the unique closed point of $S^{'} = Spec (A^{'})$ , which is the unique point of $S^{'}$ above $s$ ; if $f^{'} = f_{(S^{'})} : X^{'} \to S^{'}$ , the fibre $f^{' - 1} (s^{'})$ is canonically isomorphic to $f^{- 1} (s)$ , since $A$ and Â have the same residue field (I, 3.6.4); there is thus a single point $x^{'} \in X^{'}$ above $x$ and above $s^{'}$ , and one consequently has $O_{X^{'}, x^{'}} = O_{X, x} \otimes_{A} A^{'}$ . One deduces that the local rings $O_{X, x}$ and $O_{X^{'}, x^{'}}$ have isomorphic completions, for in general, if $E$ is a local ring which is an $A$ -algebra (the homomorphism $A \to E$ being local), the separated completion $(E \otimes_{A} \hat{A})^{\land}$ of the ring $E \otimes_{A} \hat{A}$ endowed with the tensor product topology is isomorphic to the separated completion of $\hat{E} \otimes_{A} \hat{A} = \hat{E}$ , hence to the separated completion Ê of $E$ $(0_{I}, 7.7.1)$ . By virtue of definition (19.3.1), it therefore amounts to the same to say that $O_{X, x}$ is a complete intersection ring, or that $O_{X^{'}, x^{'}}$ is a complete intersection ring.

Suppose then $A$ complete; one may moreover suppose that $X = Spec (B)$ , $B$ being a quotient of a polynomial algebra $C = A [T_{1}, \dots, T_{r}]$ . As polynomial rings over a field are regular rings (0, 17.3.7), it follows from (19.3.7) that the immersion $X \to Z = Spec (C)$ may be supposed transversally regular relatively to $S$ ; so one may suppose that $B = C / J$ , where $J$ is generated by a regular sequence $(g_{i})_{1 ⩽ i ⩽ n}$ of elements of $C$ . On the other hand, since $A$ is complete, one may by hypothesis write it $A^{'} / K$ , where $A^{'}$ is a regular local ring and $K$ an ideal of $A^{'}$ (0, 19.8.8), and the hypothesis that $A$ is a complete intersection ring entails that $K$ is generated by a regular sequence $(f_{j})_{1 ⩽ j ⩽ m}$ (19.3.2). In the polynomial ring $C^{'} = A^{'} [T_{1}, \dots, T_{r}]$ , the elements $f_{j}$ $(1 ⩽ j ⩽ m)$ still form a regular sequence generating the ideal $K^{'} = K [T_{1}, \dots, T_{r}]$ (0, 15.1.4); as $C^{'} / K^{'} = C$ , one sees that if, for each $i$ , $g_{i}^{'}$ is an element of $C^{'}$ of image $g_{i}$ in $C$ , the sequence formed of the $f_{j}$ $(1 ⩽ j ⩽ m)$ and of the $g_{i}^{'}$ $(1 ⩽ i ⩽ n)$ is regular in $C^{'}$ ; if $J^{'}$ is the ideal of $C^{'}$ it generates, one has $B = C^{'} / J^{'}$ , whence the conclusion by virtue of (19.3.2), since $C^{'}$ is a regular ring (0, 17.3.7).

19.4. Application: regularity and smoothness criteria for blow-up preschemes

(19.4.1)

Let $X$ be a prescheme, $Y$ a closed sub-prescheme of $X$ defined by a quasi-coherent Ideal $I$ of $O_{X}$ . Recall (II, 8.1.3) that the $X$ -scheme $X^{'}$ obtained by blowing up $Y$ is the prescheme

  X' = Proj(𝒮),    where    𝒮 = ⊕_{n ⩾ 0} 𝓘^n.

If $I$ is of finite type, the structure morphism $f : X^{'} \to X$ is projective. Without hypothesis on $I$ , the closed sub-prescheme

$Y^{'} = f^{- 1} (Y)$

of $X^{'}$ is defined by the quasi-coherent Ideal $I O_{X^{'}}$ of $O_{X^{'}}$ , canonically isomorphic to $O_{X^{'}} (1)$ ; more precisely, one has an exact sequence

  (19.4.1.1)    0 → 𝓘 𝒪_{X'} → 𝒪_{X'} → 𝒪_{Y'} → 0

where $I O_{X^{'}}$ is the image of $I \otimes_{O_{X}} O_{X^{'}}$ (II, 8.1.7 and 8.1.8). As in a neighbourhood of a point of $X^{'}$ , the $O_{X^{'}}$ -Module $O_{X^{'}} (1)$ is free of rank 1, $I O_{X^{'}}$ is generated in such a neighbourhood by an invertible section of $O_{X^{'}}$ , and $I O_{X^{'}} / I^{2} O_{X^{'}}$ is consequently a free $(O_{X^{'}} / I O_{X^{'}})$ -Module of rank 1. This proves the first assertion of the following lemma:

Lemma (19.4.2).

The canonical immersion $Y^{'} \to X^{'}$ is regular and of codimension 1, and $Y^{'}$ is canonically isomorphic to the $Y$ -scheme $Proj (g r_{I}^{∙} (O_{X}))$ .

The second assertion follows from (II, 3.5.3) applied to the canonical injection $Y \to X$ , taking into account that $S \otimes_{O_{X}} (O_{X} / I) = g r_{I}^{∙} (O_{X})$ by definition, since $I \otimes_{O_{X}} (O_{X} / I)$ is isomorphic to $O_{X} / I$ and $I^{n} \otimes_{O_{X}} (O_{X} / I)$ to $I^{n} / I^{n + 1}$ .

Corollary (19.4.3).

If the canonical immersion $Y \to X$ is quasi-regular, the restriction $g : Y^{'} \to Y$ of the morphism $f$ is smooth.

Indeed, $N_{Y / X} = I / I^{2}$ is then a locally free $O_{Y}$ -Module of finite rank, and $g r_{I}^{∙} (O_{X})$ is isomorphic to $S_{O_{Y}}^{∙} (N_{Y / X})$ (16.9.8), so $Y^{'}$ is $Y$ -isomorphic to $P (N_{Y / X})$ and consequently is smooth over $Y$ (17.3.9).

We shall see later (chap. V) that $g$ is still smooth when the immersion $Y \to X$ "presents only ordinary singularities".

Proposition (19.4.4).

With the notations of (19.4.1), suppose that $X$ is locally Noetherian and that the morphism $g : Y^{'} \to Y$ , restriction of $f$ , is smooth. Let $x^{'}$ be a point of $Y^{'}$ , $x = g (x^{'})$ . For $Y$ to be regular at the point $x$ , it is necessary and sufficient that $Y^{'}$ be regular at the point $x^{'}$ ; when this holds, $X^{'}$ is then regular at the point $x^{'}$ .

The first assertion follows from (17.5.8) and the second from (19.1.1) and (19.4.2).

Corollary (19.4.5).

Under the general hypotheses of (19.4.4), suppose that $Y$ is regular at the point $x$ . Then, for every generization $x_{1}$ of $x$ in $X$ not belonging to $Y$ , $X$ is regular at the point $x_{1}$ . If $R e g (X)$ is open ((6.12), for example if $X$ is excellent (7.8.6, (iii))),

and if $Y$ is regular, then there exists an open neighbourhood $U$ of $Y$ in $X$ such that $X$ is regular at the points of $U - Y$ .

To prove the first assertion, one may restrict to the case where $X$ is a local scheme of closed point $x$ ; the hypothesis then entails that $Y$ is regular (0, 17.3.2), so the same holds for $Y^{'}$ by (19.4.4). Note now that the morphism $f : X^{'} \to X$ is projective, hence proper, and therefore (II, 7.2.1), if one considers the unique point $x_{1}^{'}$ of $X^{'}$ above $x_{1}$ (II, 8.1.3), there exists a specialization of $x_{1}^{'}$ belonging to $Y^{'}$ ; applying (19.4.4) and using (0, 17.3.2) one concludes that $X^{'}$ is regular at the point $x_{1}^{'}$ , hence $X$ regular at the point $x_{1}$ since $X^{'} - Y^{'}$ is isomorphic to $X - Y$ (II, 8.1.3). If $Y$ is regular, every point of $X - Y$ which is a generization of a point of $Y$ therefore belongs to $R e g (X)$ ; if $R e g (X)$ is open, $U = Y \cup R e g (X)$ is therefore locally constructible and stable under generization, hence open $(0_{III}, 9.2.5)$ and answers the question.

Proposition (19.4.6).

Let $g : X \to S$ be a morphism, $Y$ a closed sub-prescheme of $X$ defined by a quasi-coherent Ideal $I$ , $X^{'}$ the $X$ -scheme obtained by blowing up $Y$ , $Y^{'}$ the inverse image of $Y$ in $X^{'}$ .

(i) The following conditions are equivalent:

a) $g r_{I}^{∙} (O_{X})$ is a $g$ -flat $O_{X}$ -Algebra.

b) For every $n ⩾ 0$ , the $O_{X}$ -Module $O_{X} / I^{n + 1}$ is $g$ -flat (in other words, the $n$ -th infinitesimal neighbourhood $Y_{n}$ of $Y$ in $X$ (16.1.2) is $S$ -flat).

c) For every base change $S_{1} \to S$ and every integer $n ⩾ 1$ , if one sets $X_{1} = X \times_{S} S_{1}$ , the canonical homomorphism $I^{n} \otimes_{O_{S}} O_{S_{1}} \to I_{1}^{n}$ is bijective.

When this holds, $Y^{'}$ is $S$ -flat, and for every base change $S_{1} \to S$ , if Y_1 is the inverse image of $Y$ in X_1, the prescheme $X_{1}^{'}$ obtained by blowing up Y_1 in X_1 is canonically isomorphic to $X^{'} \times_{S} S_{1}$ .

(ii) Suppose that the equivalent conditions of (i) are satisfied and moreover that the morphisms $X \to S$ and $Y \to S$ are locally of finite presentation. Then $S = \oplus_{n ⩾ 0} I^{n}$ is an $O_{X}$ -Algebra of finite presentation, the morphism $X^{'} \to X$ is of finite presentation, the canonical immersion $Y^{'} \to X^{'}$ is transversally regular of codimension 1 relatively to $S$ , and the set of points of $X$ (resp. $X^{'}$ ) where $X$ (resp. $X^{'}$ ) is $S$ -flat is a neighbourhood of $Y$ (resp. $Y^{'}$ ).

(i) The equivalence of a) and b) follows at once from $(0_{I}, 6.1.2)$ ; to show the equivalence of b) and c), one may restrict to the case where $S = Spec (A)$ and $X = Spec (B)$ are affine, with $I = \tilde{J}$ , where $J$ is an ideal of $B$ ; the exact sequence of Tor then shows that condition b) implies that $T o r_{1}^{A} (B / J^{n + 1}, A^{'}) = 0$ for every integer $n ⩾ 0$ and every $A$ -algebra $A^{'}$ ; by taking for $A^{'}$ an algebra direct sum $A \oplus M$ , where $M$ is an arbitrary $A$ -module and where multiplication is defined by $(a, m) (a^{'}, m^{'}) = (a a^{'}, a m^{'} + a^{'} m)$ , one sees that the preceding condition is equivalent to $T o r_{1}^{A} (B / J^{n + 1}, M) = 0$ , hence to the fact that $B / J^{n + 1}$ is a flat $A$ -module. If one sets $J_{1} = J \otimes_{A} A_{1}$ , one then has $\oplus_{n ⩾ 0} J_{1}^{n} = (\oplus_{n ⩾ 0} J^{n}) \otimes_{A} A_{1}$

and the assertion relative to the blown-up prescheme $X_{1}^{'}$ follows at once. Finally, to prove that $Y^{'}$ is $S$ -flat, one may again restrict to the case where $S$ and $X$ are affine; if one sets $C = g r_{I}^{∙} (B)$ , every point of $Y^{'}$ then admits, by virtue of (19.4.2) and (II, 2.3.6), an affine open neighbourhood whose ring is of the form $C_{t}$ , where $t$ is a homogeneous element of degree 1 of $C$ ; since by hypothesis $C$ is a flat $A$ -module, so is its ring of fractions $C_{t}$ and the degree-0 component $C_{t}$ of this graded $A$ -module, whence our assertion.

(ii) One may always restrict to the case where $S$ and $X$ are affine, $B$ being therefore an $A$ -algebra of finite presentation, $J$ an ideal of finite type of $B$ . By virtue of (8.9.1), (8.6.3) and (11.2.9), there exist a Noetherian sub-ring A_0 of $A$ , an A_0-algebra of finite type B_0, an ideal $J_{0}$ of B_0 such that $B = B_{0} \otimes_{A_{0}} A$ , $J = J_{0} \otimes_{A_{0}} A$ , such that $g r_{J_{0}}^{∙} (B_{0})$ is an A_0-flat module, and that $g r_{J}^{∙} (B) = g r_{J_{0}}^{∙} (B_{0}) \otimes_{A_{0}} A$ . By virtue of (i), one then has $\oplus_{n ⩾ 0} J^{n} = (\oplus_{n ⩾ 0} J_{0}^{n}) \otimes_{A_{0}} A = (\oplus_{n ⩾ 0} J_{0}^{n}) \otimes_{B_{0}} B$ ; as $\oplus_{n ⩾ 0} J_{0}^{n}$ is a B_0-algebra generated by the ideal of finite type $J_{0}$ of B_0, hence a B_0-algebra of finite type, and consequently of finite presentation since B_0 is Noetherian, one deduces that $\oplus_{n ⩾ 0} J^{n}$ is a $B$ -algebra of finite presentation. Likewise, the morphism $Proj (\oplus_{n ⩾ 0} J_{0}^{n}) \to Spec (B_{0})$ is of finite type (II, 2.7.1), and consequently of finite presentation; so the morphism $X^{'} \to X$ , which is deduced from it by base change, is of finite presentation. One already knows (19.4.2) that the canonical immersion $Y^{'} \to X^{'}$ is regular; as $Y^{'}$ is $S$ -flat, as one saw in (i), one deduces from (19.2.4) that the immersion $Y^{'} \to X^{'}$ is transversally regular relatively to $S$ and that $X^{'}$ is $S$ -flat at the points of $Y^{'}$ , hence (11.3.1) in an open neighbourhood of $Y^{'}$ . On the other hand, the fact that $g r_{I}^{∙} (O_{X})$ is an $S$ -flat $O_{X}$ -Algebra entails, by (11.3.4), that $X$ is $S$ -flat at the points of $Y$ , hence (11.3.1) in a neighbourhood of $Y$ .

Corollary (19.4.7).

Let $g : X \to S$ be a morphism locally of finite presentation, $Y$ a closed sub-prescheme of $X$ such that the composite morphism $Y \to X \to S$ is locally of finite presentation; suppose moreover that $Y$ is $S$ -flat and that $O_{X}$ is normally flat along $Y$ (11.3.4). Then (with the notations of (19.4.1)), the morphism $X^{'} \to X$ is of finite presentation, the canonical immersion $Y^{'} \to X^{'}$ is transversally regular of codimension 1 relatively to $S$ , $Y^{'}$ is $Y$ -flat (hence $S$ -flat), and the set of points of $X$ (resp. $X^{'}$ ) where $X$ (resp. $X^{'}$ ) is $S$ -flat is a neighbourhood of $Y$ (resp. $Y^{'}$ ).

Indeed, by hypothesis $g r_{I}^{∙} (O_{X})$ is a flat $(O_{X} / I)$ -Module, hence is $S$ -flat since $Y$ is supposed to be $S$ -flat. One may consequently apply the results of (19.4.6, (i) and (ii)). As $Y^{'}$ is $Y$ -isomorphic to $Proj (g r_{I}^{∙} (O_{X}))$ (19.4.2), the same reasoning as in (19.4.6) shows that $Y^{'}$ is $Y$ -flat.

Proposition (19.4.8).

Suppose the hypotheses of (19.4.7) are verified and moreover that the morphism $Y^{'} \to Y$ is smooth. Let $x^{'}$ be a point of $Y^{'}$ , $x$ its image in $Y$ . For $Y$ to be smooth over $S$ at the point $x$ , it is necessary and sufficient that $Y^{'}$ be smooth over $S$ at the point $x^{'}$ ; when this holds, $X^{'}$ is then smooth over $S$ at the point $x^{'}$ .

The preceding conclusions are in particular verified when the morphism $X \to S$ is locally of finite presentation and the immersion $Y \to X$ is transversally regular relatively to $S$ .

Taking into account the compatibility of the formation of a blow-up prescheme with base changes in the case considered (19.4.6, (i)) and (17.5.1) and (17.7.1),

it suffices to consider the case where $S$ is the spectrum of an algebraically closed field. But then it amounts to the same to say that an $S$ -prescheme locally of finite type is smooth over $S$ at a point or that it is regular at this point (17.15.1), so the conclusions follow from (19.4.4).

When $X \to S$ is locally of finite presentation and the canonical immersion $Y \to X$ is quasi-regular, this immersion is by definition of finite presentation since the Ideal which defines $Y$ is of finite type, so the composite morphism $Y \to X \to S$ is locally of finite presentation. One has already seen (19.4.3) that under these conditions the morphism $Y^{'} \to Y$ is smooth; moreover, one then knows that $N_{Y / X}$ is a locally free $O_{Y}$ -Module and that $g r_{I}^{∙} (O_{X})$ is isomorphic to $S_{O_{Y}}^{∙} (N_{Y / X})$ , hence is a flat $O_{Y}$ -Module. The hypotheses of (19.4.7) are therefore all verified and one may apply the conclusions of the statement of (19.4.8).

Corollary (19.4.9).

Under the general hypotheses of (19.4.8), suppose moreover that $Y$ is smooth over $S$ at the point $x$ . Then, for every generization $x_{1}$ of $x$ in $X$ not belonging to $Y$ , $X$ is smooth over $S$ at the point $x_{1}$ . If $Y$ is smooth over $S$ , then there exists an open neighbourhood $U$ of $Y$ in $X$ such that $X$ is smooth over $S$ at the points of $U - Y$ .

For the first assertion, one may restrict to the case where $X$ is a local scheme of closed point $x$ , and the hypothesis then entails that $Y$ is smooth over $S$ , every neighbourhood of the closed point of $Y$ in $Y$ being necessarily $Y$ itself; one therefore concludes by the same reasoning as in (19.4.5), using the fact that the set of points where $X^{'}$ is smooth over $S$ is open in $X^{'}$ . To prove the second assertion, one may reduce to the case where $S$ is Noetherian, and $X$ of finite type over $S$ , thanks to (8.9.1), (8.6.3), (11.2.9) and (17.7.6); one then concludes as in (19.4.5), using the fact that the set of points where $X$ is smooth over $S$ is open in $X$ .

Remark (19.4.10).

In (19.4.6, (ii)), replace the hypothesis that $X$ and $Y$ are locally of finite presentation over $S$ by the hypothesis that $S$ and $X$ (hence also $Y$ ) are locally Noetherian. Then it is clear that $\oplus_{n ⩾ 0} I^{n}$ is an $O_{X}$ -Algebra of finite type, that the morphism $X^{'} \to X$ is of finite type, and it still follows from (19.2.4) applied to locally Noetherian preschemes that the immersion $Y^{'} \to X^{'}$ is transversally regular relatively to $S$ and that $X^{'}$ is $S$ -flat at the points of $Y^{'}$ .

Proposition (19.4.11).

Let $X$ be a prescheme, $E$ a quasi-coherent $O_{X}$ -Module, $u : E \to O_{X}$ a homomorphism of $O_{X}$ -Modules, $I = u (E)$ the quasi-coherent Ideal of $O_{X}$ image of $u$ , $Y$ the closed sub-prescheme of $X$ defined by $I$ . Let $X^{'}$ be the $X$ -scheme obtained by blowing up $Y$ . On the other hand, let $P = P (E)$ be the projective bundle on $X$ defined by $E$ (II, 4.1.1), $p : P \to X$ the structure morphism, $α ♯ : p * (E) \to O_{P} (1)$ the canonical homomorphism (II, 4.1.5.1) and $H$ its kernel, so that one has the exact sequence

  0 → ℋ → p*(ℰ) →^{α♯} 𝒪_P(1) → 0.

Finally, let $J$ be the quasi-coherent Ideal of $O_{P}$ image of the restriction $v : H \to O_{P}$ of $p * (u)$ , and let $Z$ be the closed sub-prescheme of $P$ defined by $J$ .

(i) The image of $X^{'}$ by the closed immersion $j : X^{'} \to P$ corresponding to the surjective homomorphism of graded $O_{X}$ -Algebras $S_{O_{X}}^{∙} (E) \to S_{O_{X}}^{∙} (I) \to \oplus_{n ⩾ 0} I^{n}$ (II, 3.6.2) is a closed sub-prescheme $j (X^{'})$ of $Z$ .

(ii) If $E$ is a locally free $O_{X}$ -Module of rank $m$ , $H$ is a locally free $O_{P}$ -Module of rank $m - 1$ .

(iii) Suppose that $X$ is locally Noetherian, that $E$ is locally free of finite type, that the sub-prescheme $Y$ is regular, and that at every point $x \in Y$ , the canonical immersion $i : Y \to X$ is regular and of codimension equal to $r g_{O_{x}} (E_{x})$ (which we shall express by saying that the homomorphism $u$ is regular at the point $x$ (cf. chap. V)). Then the image of $X^{'}$ by the closed immersion $j : X^{'} \to P$ is the closed sub-prescheme $Z$ of $P$ ; at all points of $X^{'} - Y^{'}$ (with $Y^{'}$ the inverse image of $Y$ in $X^{'}$ ) the immersion $j$ is transversally regular relatively to $X$ ; finally at all points $x^{'}$ of $Y^{'}$ , $X^{'}$ is regular, the immersion $j$ is regular and of codimension $r g_{O_{z}} (H_{z})$ where $z = j (x^{'})$ (in other words the homomorphism $v : H \to O_{P}$ is regular at $z$ ).

All questions being local on $X$ , one may restrict to the case where $X = Spec (A)$ is affine and $E = \tilde{E}$ , where $E$ is an $A$ -module. To do the verification of (i), one may moreover restrict to examining what happens in an affine open of $P$ of the form $D_{+} (t)$ , where $t \in E = S_{1} (E)$ (II, 2.3.14). If one sets $S = S_{A}^{∙} (E)$ , one then has $Γ (D_{+} (t), O_{P}) = S_{t}$ (II, 2.4.1), and $Γ (D_{+} (t), p * (E)) = (E \otimes_{A} S_{t})$ . If one refers to the definition of $α ♯$ (II, 2.6.2.2), one sees that to every element $a = \sum_{i} x_{i}^{(1)} \otimes ((x_{i}^{(2)} \dots x_{i}^{(n + 1)}) / t^{n})$ of $E \otimes_{A} S_{t}$ (where $x_{i}^{(1)}$ and the $x_{i}^{(j)}$ are in $E$ ) $α ♯$ makes correspond the element $\sum_{i} (x_{i}^{(1)} x_{i}^{(2)} \dots x_{i}^{(n + 1)}) / t^{n + 1}$ of $(S (1))_{t}$ . It is a matter (to prove (i)) of showing that if this latter element is zero, then so is the image of $a^{'} = v (a) = \sum_{i} u (x_{i}^{(1)}) ((x_{i}^{(2)} \dots x_{i}^{(n + 1)}) / t^{n})$ by the canonical homomorphism S_(t) → (𝓘 ⊗_𝒪_X (𝒪_X/𝓘))_t (II, 3.6.2). Now, this image is none other than $\sum_{i} u (x_{i}^{(1)}) u (x_{i}^{(2)}) \dots u (x_{i}^{(n + 1)}) / (u (t))^{n + 1}$ in the ring $A_{u (t)}$ , that is, the product of the element $u (t)$ and the canonical image of $\sum_{i} u (x_{i}^{(1)}) (x_{i}^{(2)} \dots x_{i}^{(n + 1)}) / t^{n + 1}$ by the algebra homomorphism, canonical extension to $S_{t}$ of the homomorphism $u : E \to A$ of $A$ -modules. This therefore proves (i).

To establish (ii), let us note that one may suppose that $E$ is a free $A$ -module of rank $m$ ; since $α ♯ : E \otimes_{A} S (1) \to S (1)$ is surjective and $(S (1))_{t}$ is a free $S_{t}$ -module of rank 1, the exact sequence

  0 → H_(t) → (E ⊗_A S(1))_(t) → (S(1))_(t) → 0

is split, and $H_{t}$ is therefore a projective $S_{t}$ -module of rank $m - 1$ , whence (ii).

To prove (iii), let us first examine what happens above a point $x \in X - Y$ . Then $u : E \to O_{X}$ restricted to a neighbourhood of $x$ , is surjective, so one may restrict to the case where $E = O_{X}$ , in which case $X^{'}$ is canonically identified with $X$ . To see then that the closed sub-prescheme of $P$ image of $X^{'}$ is identified with $Z$ , one may restrict as at the beginning to the case where $X = Spec (A)$ is affine, and moreover $E = A^{m}$ , so that $S = A [T_{0}, \dots, T_{m - 1}]$ . One may moreover, with the notations of the beginning, suppose $u (t) \neq = 0$ (the values of $t \in E$

verifying this condition generate the symmetric algebra $S_{A}^{∙} (E)$ ); by changing if necessary the basis in $E$ , one may therefore suppose that $t = T_{0}$ , and $u (T_{i}) = 0$ for $1 ⩽ i ⩽ m - 1$ , so that $S_{t}$ is canonically identified with $A [T_{1}, \dots, T_{m - 1}]$ . The elements of $H$ are then the elements of the form $\sum_{α} L_{α} T_{0} T^{α}$ of $E \otimes_{A} A [T_{1}, \dots, T_{m - 1}]$ where the $L_{α} = λ_{α, 0} + λ_{α, 1} T_{1} + \dots + λ_{α, m - 1} T_{m - 1}$ must be such that $\sum_{α} L_{α} T^{α} = 0$ in $A [T_{1}, \dots, T_{m - 1}]$ . One verifies easily that when the condition $λ_{α, 0} = 0$ is fulfilled, the values of the $λ_{α, j}$ may be taken arbitrarily for $∣ α ∣ ⩾ 1$ , and one may always then determine the $λ_{α, j}$ for $j ⩾ 1$ so as to verify the condition $\sum_{α} L_{α} T^{α} = 0$ . The elements of the ideal $J$ corresponding are then the polynomials $\sum_{α} λ_{α} T^{α}$ without constant term, and since $(S_{A}^{∙} (A))_{t}$ is identified with $A$ , these polynomials are exactly the elements of the kernel of the homomorphism $(S_{A}^{∙} (E))_{t} \to (S_{A}^{∙} (A))_{t}$ . One has thus proved (19.2.1) that the immersion $j : X^{'} \to P$ is transversally regular relatively to $X$ at all points of $X^{'} - Y^{'}$ and that $j (X^{'}) = Z$ at these points.

It remains to examine what happens above a point $x \in Y$ . The hypothesis of "regularity" of $u$ at this point made in (iii) is equivalent (by virtue of (19.1.1) and (0, 17.1.7)) to saying that $X$ is regular at the point $x$ and that the homomorphism $u_{x} \otimes 1 : E_{x} \otimes k (x) \to m_{x} / m_{x}^{2}$ is injective. This being so, as the morphism $p : P \to X$ is smooth (17.3.9), for every $z \in P$ above $x$ , $P$ is regular at the point $z$ (17.5.8); furthermore (0, 17.3.3) the canonical homomorphism

  (19.4.11.1)    (𝔪_x/𝔪_x²) ⊗_{k(x)} k(z) → 𝔪_z/𝔪_z²

is injective. One deduces that the homomorphism

  (ℰ_x ⊗_{𝒪_{X, x}} 𝒪_{P, z}) ⊗_{𝒪_{P, z}} k(z) → 𝔪_z/𝔪_z²

deduced from $p * (u)$ is also injective, for it is written as the composite

  (ℰ_x ⊗_{𝒪_{X, x}} k(z)) → (𝔪_x/𝔪_x²) ⊗_{k(x)} k(z) → 𝔪_z/𝔪_z²

where the first arrow is injective by flatness and the second is the injective homomorphism (19.4.11.1). As $H$ is (in a neighbourhood of $z$ in $P$ ) a direct factor of $E \otimes_{O_{X}} O_{P} = p * (E)$ , the homomorphism $v_{z} \otimes 1 : H_{z} \otimes_{O_{P, z}} k (z) \to m_{z} / m_{z}^{2}$ is also injective, which establishes the "regularity" of the homomorphism $v$ at the point $z$ . It remains to see that the image by $j$ of $X^{'}$ is identical to the closed sub-prescheme $Z$ . It will suffice for this to show on the one hand that the image by $j$ of the points of $q^{- 1} (x)$ (where $q : X^{'} \to X$ is the structure morphism) are exactly those of $Z \cap p^{- 1} (x)$ and on the other hand that at each of these points $z = j (x^{'})$ , the surjective homomorphism $O_{Z, z} \to O_{X^{'}, x^{'}}$ is bijective. But, taking into account the hypothesis and (19.4.3), the restriction $Y^{'} \to Y$ of $q$ is smooth, and since $O_{Y, x}$ is supposed regular, the same holds for $O_{Y^{'}, x^{'}}$ (17.5.8); furthermore, the immersion $Y^{'} \to X^{'}$ is regular (19.4.2), so $O_{X^{'}, x^{'}}$ is also a regular ring (19.1.1). To prove that the homomorphism $O_{Z, z} \to O_{X^{'}, x^{'}}$ is bijective, it suffices then to show that the dimensions

of these two regular rings are equal (0, 17.1.9). But by virtue of (0, 17.3.3), this is also equivalent to the equality of the dimensions of the two rings $O_{X^{'}, x^{'}} \otimes_{O_{X}} k (x)$ and $O_{Z, z} \otimes_{O_{X}} k (x)$ , so that finally one may replace $X^{'}$ and $Z$ by the fibres $q^{- 1} (x)$ and $Z \cap p^{- 1} (x)$ respectively, and reduce to proving that the morphism $q^{- 1} (x) \to Z \cap p^{- 1} (x)$ deduced from $j$ is an isomorphism. Now, one may restrict here to the case where $X = Spec (B)$ with $B = O_{X, x}$ and $E = B^{m}$ , whence $S = B [T_{1}, \dots, T_{m}]$ ; furthermore, the hypothesis means that there exists a regular system of parameters $(t_{i})_{1 ⩽ i ⩽ m}$ of $B$ such that $u (T_{i}) = t_{i}$ for $1 ⩽ i ⩽ m$ (0, 17.1.7); let $J$ be the image of $u$ ; if $m$ is the maximal ideal of $B$ , one has $J^{n} \otimes_{B} (B / m) = J^{n} / m J^{n} = (J^{n} / J^{n + 1}) \otimes_{B} (B / m)$ and consequently (16.9.4.1), $(\oplus_{n ⩾ 0} J^{n}) \otimes_{B} k (x)$ is identified with $k (x) [T_{1}, \dots, T_{m}]$ , whence it follows that the immersion $q^{- 1} (x) \to p^{- 1} (x)$ is here an isomorphism; this implies a fortiori that the closed sub-prescheme $Z \cap p^{- 1} (x)$ is identical to the fibre $j (q^{- 1} (x))$ , and terminates the proof of (19.4.11).

19.5. $M$ -regularity criteria

We resume here, in completed form, the criteria for a sequence of elements of a ring $A$ to be $M$ -regular or $M$ -quasi-regular ( $M$ being an $A$ -module), already studied in (0, 15.1).

Theorem (19.5.1).

Let $A$ be a ring, $f = (f_{i})_{1 ⩽ i ⩽ n}$ a sequence of elements of $A$ , $M$ an $A$ -module. Consider the following properties:

a) The sequence $f$ is $M$ -regular.

b) One has $H_{i} (f, M) = 0$ for every $i > 0$ (III, 1.1.3).

b') One has $H_{1} (f, M) = 0$ .

c) The sequence $f$ is $M$ -quasi-regular.

Then one has the implications

  a) ⟹ b) ⟹ b')    and    a) ⟹ c).

Set $J = \sum_{i = 1}^{n} f_{i} A$ . If the modules $M_{i} = M / (\sum_{j ⩽ i} f_{j} M)$ are separated for the $J$ -preadic topology (resp. if every quotient module of a sub-module of $M$ is separated for the $J$ -preadic topology), then one also has the implication c) ⟹ a) (resp. b') ⟹ a)).

The implications a) ⟹ c), and c) ⟹ a) when the $M_{i}$ are separated for the $J$ -preadic topology, were already proved (0, 15.1.9), and are given only for the record. One has also shown (III, 1.1.4 and 1.1.3.3) that a) entails b), and b) trivially implies b'). It therefore remains to see that when every quotient module of a sub-module of $M$ is separated for the $J$ -preadic topology, b') entails a). Let us reason by induction on $n$ . For $n = 1$ , the assertion follows trivially from the definitions, since $H_{1} (f, M)$ is none other than the kernel of the homothety $x \mapsto f_{1} x$ in $M$ (III, 1.1.1 and 1.1.2). Suppose then $n ⩾ 2$ , and set $f^{'} = (f_{i})_{1 ⩽ i ⩽ n - 1}$ ; with the notations of (III, 1.1.2), one has $K_{∙} (f, M) = K_{∙} (f_{n}) \otimes_{A} K_{∙} (f^{'}, M)$ . One therefore has (III, 1.1.4.1) the exact sequence

  (19.5.1.1)    0 → H_0(f_n, H_1(f', M)) → H_1(f, M) → H_1(f_n, H_0(f', M)) → 0.

The relation $H_{1} (f, M) = 0$ therefore implies $H_{0} (f_{n}, H_{1} (f^{'}, M)) = 0$ and $H_{1} (f_{n}, H_{0} (f^{'}, M)) = 0$ . The first of these two relations means that one has $f_{n} H_{1} (f^{'}, M) = H_{1} (f^{'}, M)$ (III, 1.1.3.5). Now, by definition (III, 1.1.1), $H_{1} (f^{'}, M)$ is isomorphic to a quotient of a sub-module $N$ of $M^{n - 1}$ ; taking on $M^{n - 1}$ the filtration of the $M^{j}$ for $j ⩽ n - 1$ , on $N$ the induced filtration, and on $H_{1} (f^{'}, M)$ the quotient filtration of that of $N$ , one obtains on $H_{1} (f^{'}, M)$ a finite filtration whose quotients are isomorphic to quotients of sub-modules of $M$ , hence are separated for the $J$ -preadic topology by hypothesis. As the relation $f_{n} H_{1} (f^{'}, M) = H_{1} (f^{'}, M)$ implies a fortiori $J H_{1} (f^{'}, M) = H_{1} (f^{'}, M)$ , one deduces from the hypothesis and the preceding remarks that $H_{1} (f^{'}, M) = 0$ . The induction hypothesis then proves that the sequence $f^{'}$ is $M$ -regular. On the other hand, the relation $H_{1} (f_{n}, H_{0} (f^{'}, M)) = 0$ means that $f_{n}$ is $(M / (\sum_{i = 1}^{n - 1} f_{i} M))$ -regular, so the sequence $f$ is $M$ -regular.

Corollary (19.5.2).

Suppose that $A$ is a Noetherian ring, that the $f_{i}$ $(1 ⩽ i ⩽ n)$ belong to the radical of $A$ , and that $M$ is an $A$ -module of finite type. Then the four conditions a), b), b'), c) of (19.5.1) are equivalent.

One knows indeed $(0_{I}, 7.3.5)$ that every $A$ -module of finite type is separated for the $J$ -preadic topology.

Proposition (19.5.3).

Let

$0 \to M^{'} \to M \to M^{''} \to 0$

be an exact sequence of $A$ -modules, $f = (f_{i})_{1 ⩽ i ⩽ n}$ an $M$ -regular sequence, $J = \sum_{i = 1}^{n} f_{i} A$ . Consider the following properties:

a) The sequence $f$ is M''-regular.

b) The $J$ -preadic filtration of $M^{'}$ is induced on $M^{'}$ by the $J$ -preadic filtration of $M$ (in other words, the canonical homomorphism $g r_{J}^{∙} (M^{'}) \to g r_{J}^{∙} (M)$ is injective).

c) The canonical homomorphism $M^{'} / (\sum_{i = 1}^{n} f_{i} M^{'}) \to M / (\sum_{i = 1}^{n} f_{i} M)$ is injective.

Then one has the implications a) ⟹ b) ⟹ c), and a) entails moreover that the sequence $f$ is $M^{'}$ -regular.

If every quotient of a sub-module of M'' is separated for the $J$ -preadic topology, the conditions a), b) and c) are equivalent.

It is clear that c) is a consequence of b). Let us prove that under the hypotheses of the last assertion, c) entails a): since $f$ is $M$ -regular, one has, by (19.5.1), $H_{1} (f, M) = 0$ , whence an exact sequence, portion of the exact homology sequence

  0 → H_1(f, M'') → H_0(f, M') → H_0(f, M).

Now, condition c) expresses that the homomorphism $H_{0} (f, M^{'}) \to H_{0} (f, M)$ is injective (III, 1.1.3.5); it therefore amounts to $H_{1} (f, M^{''}) = 0$ , whence, by virtue of the separation hypothesis and (19.5.1), the fact that $f$ is M''-regular.

Let us next show that a) entails c) and the fact that $f$ is $M^{'}$ -regular, by induction on $n$ . For $n = 1$ , $f_{1}$ being $M$ -regular is also $M^{'}$ -regular and property c) is none other than lemma (3.4.1.4). For $n > 1$ , the induction hypothesis shows that

if one sets $f^{'} = (f_{1}, \dots, f_{n - 1})$ , the sequence $f^{'}$ is $M^{'}$ -regular and one has, by virtue of c) applied to $f^{'}$ , an exact sequence

  0 → M'/(∑_{i=1}^{n−1} f_i M') → M/(∑_{i=1}^{n−1} f_i M) → M''/(∑_{i=1}^{n−1} f_i M'') → 0.

By hypothesis, $f_{n}$ is $(M / (\sum_{i = 1}^{n - 1} f_{i} M))$ -regular and $(M^{''} / (\sum_{i = 1}^{n - 1} f_{i} M^{''}))$ -regular, so the same reasoning shows on the one hand that $f_{n}$ is $(M^{'} / (\sum_{i = 1}^{n - 1} f_{i} M^{'}))$ -regular, and on the other hand, by virtue of (3.4.1.4), that the sequence

  0 → M'/(∑_{i=1}^n f_i M') → M/(∑_{i=1}^n f_i M) → M''/(∑_{i=1}^n f_i M'') → 0

is exact, whence c).

Let us finally show that a) entails b). As the sequence $f$ is then $M$ -regular and $M^{'}$ -regular, hence $M$ -quasi-regular and $M^{'}$ -quasi-regular, one has canonical isomorphisms

  gr_𝔍^•(M') ≃ gr_𝔍^0(M')[T_1, …, T_n],    gr_𝔍^•(M) ≃ gr_𝔍^0(M)[T_1, …, T_n]

(0, 15.1.7), and as one has seen above that $g r_{J}^{0} (M^{'}) \to g r_{J}^{0} (M)$ is injective, so is $g r_{J}^{∙} (M^{'}) \to g r_{J}^{∙} (M)$ .

One will note again that the equivalence of conditions a), b), c) of (19.5.3) is valid in particular when $A$ is Noetherian, M'' an $A$ -module of finite type and the $f_{i}$ belong to the radical of $A$ .

(19.5.4)

In the sequel of this number, we keep the notations $A$ , $f$ , $M$ , $J$ of (19.5.1), and we suppose moreover that $M$ is endowed with a decreasing filtration $(M_{k})$ formed of sub- $A$ -modules, such that $M_{0} = M$ . Recall (0, 15.1.5) that one then defines on $M$ a second decreasing filtration formed of the sub-modules

  M'_k = M_k + 𝔍 M_{k−1} + ⋯ + 𝔍^{k−1} M_1 + 𝔍^k M_0

and that, if $g r_{∙} (M)$ and $g r_{∙}^{'} (M)$ are the graded $A$ -modules associated with the filtrations $(M_{k})$ and $(M_{k}^{'})$ respectively, one defines a surjective graded homomorphism of degree 0 (0, 15.1.5.2)

  ψ_M : (gr_•(M) ⊗_A (A/𝔍))[T_1, …, T_n] → gr'_•(M).

Recall also that one has the canonical surjective homomorphism (0, 15.1.1.1)

  φ_{gr_•(M)} : (gr_•(M) ⊗_A (A/𝔍))[T_1, …, T_n] → gr_𝔍^•(gr_•(M)).

Theorem (19.5.5).

With the notations of (19.5.4), consider the following properties:

a) The sequence $f$ is $g r_{∙} (M)$ -regular.

b) The canonical homomorphism $ψ_{M}$ is bijective.

c) The canonical homomorphism $ϕ_{g r_{∙} (M)}$ is bijective (in other words (0, 15.1.7), the sequence $f$ is $g r_{∙} (M)$ -quasi-regular).

One then has the implications¹

  a) ⟹ b) ⟹ c).

Moreover, if, for every integer $k ⩾ 0$ , the quotient modules of sub-modules of $g r_{k} (M)$ are separated for the $J$ -preadic topology (which is the case when $A$ is Noetherian, the $f_{i}$ are in the radical of $A$ and the $g r_{k} (M)$ are $A$ -modules of finite type), then the conditions a), b) and c) are equivalent.

The implication a) ⟹ b) was proved in (0, 15.1.8); the implication c) ⟹ a) under the separation hypothesis is a particular case of (0, 15.1.9). It remains to prove that b) implies c), which will be done in several steps.

We shall denote by $J$ the ideal $\sum f_{i} A$ generated by the $f_{i}$ ; for every sequence $q = (q_{1}, \dots, q_{n})$ of $n$ integers $⩾ 0$ , we shall set $∣ q ∣ = \sum q_{i}$ and $f^{q} = f_{1}^{q_{1}} f_{2}^{q_{2}} \dots f_{n}^{q_{n}}$ .

(19.5.5.1)

For the map $ψ_{M}$ to be injective (and consequently bijective), it is necessary and sufficient that for $p ⩾ 0$ and $q ⩾ 0$ the following condition be verified:

(I_{q, p}) For every family $(x_{q})_{∣ q ∣ = q}$ of elements of $M_{p}$ , the relation

  ∑_{|q| = q} f^q x_q ∈ ∑_{i=1}^n 𝔍^{q−1} M_{p+1} + 𝔍^q M_{p+1}

implies $x_{q} \in M_{p + 1} + J M_{p}$ for every $q$ such that $∣ q ∣ = q$ .

One proceeds as in (0, 15.1.6) by considering the sub- $A$ -module $Q_{k}$ (resp. $Q_{k}^{'}$ ) of terms of degree $k$ in the first (resp. second) member of (0, 15.1.5.2). One endows $Q_{k}$ with the filtration

  (Q_k)_• = ⊕_{|q| ⩾ •} ((gr_{k−|q|}(M)) ⊗_A (A/𝔍)) T^q

and $Q_{k}^{'}$ with the image filtration formed by $ψ_{M} ((Q_{k})_{∙}) = (Q_{k}^{'})_{∙}$ ; it again suffices to prove that the homomorphisms $(ψ_{M})_{k} : g r_{∙} (Q_{k}) \to g r_{∙} (Q_{k}^{'})$ are injective. Now, one has

  gr_•(Q_k)_i = ⊕_{|q| = i} ((M_{k−i}/M_{k−i+1}) ⊗_A (A/𝔍)) T^q = ⊕_{|q| = i} (M_{k−i}/(𝔍 M_{k−i} + M_{k−i+1})) T^q;

on the other hand, $(Q_{k}^{'})_{i}$ is the image of $\sum_{∣ q ∣ = i} J^{∣ q ∣ - 1} M_{k - ∣ q ∣ + 1}$ in $M_{k} / M_{k + 1}^{'}$ . To write that $(ψ_{M})_{k}$ is injective therefore amounts to writing the condition $(I_{i, k - i})$ ; whence our assertion.

(19.5.5.2)

For the sequence $f$ to be $g r_{∙} (M)$ -quasi-regular, it suffices that, for $p ⩾ 0$ and $q > 0$ , the following condition be verified:

(S_{q, p}) For every family $(x_{q})_{∣ q ∣ = q}$ of elements of $M_{p}$ , the relation $\sum_{∣ q ∣ = q} f^{q} x_{q} \in M_{p + q + 1}$ implies $x_{q} \in M_{p + 1} + J M_{p}$ for every $q$ such that $∣ q ∣ = q$ .

By definition, to say that $f$ is $g r_{∙} (M)$ -quasi-regular means that, for every $p ⩾ 0$ and every $q ⩾ 0$ , a relation

  ∑_{|q| = q} f^q x_q ∈ M_{p+q+1} + 𝔍 M_{p+q}

for a family $(x_{q})_{∣ q ∣ = q}$ of elements of $M_{p}$ entails $x_{q} \in M_{p + 1} + J M_{p}$ for every $q$ such that $∣ q ∣ = q$ . Now, the hypothesis on the sequence $(x_{q})$ means that there exists a family $(y_{q})_{∣ q ∣ = q}$ of elements of $J M_{p}$ such that one has $\sum f^{q} x_{q} - y_{q} \in M_{p + q + 1}$ . Condition $(S_{q, p})$ then entails $x_{q} - y_{q} \in M_{p + 1} + J M_{p}$ for every $q$ such that $∣ q ∣ = q$ , whence $x_{q} \in M_{p + 1} + J M_{p}$ ; one therefore sees that the conditions $(S_{q, p})$ entail that the sequence $f$ is $g r_{∙} (M)$ -quasi-regular.

It therefore suffices to prove the following proposition:

(19.5.5.3)

For every $r ⩾ 0$ , if the conditions $(I_{q, p})$ are verified for $p + q < r$ , then the conditions $(S_{q, p})$ are verified for $p + q < r$ .

Let us introduce the conditions

(A_{q, p}) For every family $(x_{q})_{∣ q ∣ = q}$ of elements of $M_{p}$ , the relation $\sum_{∣ q ∣ = q} f^{q} x_{q} \in M_{p + q + 1}$ implies $\sum_{∣ q ∣ = q} f^{q} x_{q} \in \sum_{i = 1}^{n} J^{q - 1} M_{p + 1}$ .

It is clear that the conjunction of $(A_{q, p})$ and of $(I_{q, p})$ implies $(S_{q, p})$ . On the other hand, the conditions $(A_{0, p})$ and $(S_{0, p})$ are trivial. Consequently, (19.5.5.3) will result from the following proposition:

(19.5.5.4)

Whatever $p ⩾ 0$ and $q ⩾ 1$ , the condition $(A_{q, p})$ is entailed by the conjunction of the conditions $(A_{q - 1, p})$ , $(I_{q - 1, p + 1})$ , $(I_{q - 2, p + 2})$ , …, $(I_{0, p + q})$ .

Indeed, once this proposition is proved, the conditions $(I_{q, p})$ supposed verified for $p + q < r$ will entail, for each $p < r$ , the condition $(A_{q, p})$ for $1 ⩽ q < r - p$ , by induction on $q$ .

Let us therefore prove (19.5.5.4). Note that condition $(A_{q, p})$ is equivalent to

  (19.5.5.5)    𝔍^q M_p ∩ M_{p+q+1} ⊂ ∑_{i=1}^n 𝔍^{q−1} M_{p+1}.

Let therefore $m \in J^{q} M_{p} \cap M_{p + q + 1} \subset J M_{p} \cap M_{p + 1}$ . Applying the condition $(A_{0, p + 1})$ , one sees that there exists, for each $i$ such that $1 ⩽ i ⩽ q$ , a family $(y_{q}^{(i)})_{∣ q ∣ = q - i}$ of elements of $M_{p + i}$ such that

  m = ∑_{i=1}^q (∑_{|q| = q − i} f^q y_q^{(i)}).

Suppose that, for $1 ⩽ i < j$ $(j ⩾ 1)$ , one has proved that

  y_q^{(i)} ∈ 𝔍 M_{p+i} + M_{p+i+1}    (for |q| = q − i).

One deduces by definition (19.5.4)

  ∑_{|q| = q − j} f^q y_q^{(j)} ∈ M'_{p+q+1}.

Since by hypothesis $(I_{q - j, p + j})$ is verified, one deduces

  y_q^{(j)} ∈ 𝔍 M_{p+j} + M_{p+j+1}

and, by induction on $j$ , one sees therefore that this condition is verified for every $j$ such that $1 ⩽ j ⩽ q$ . One therefore has

  m ∈ ∑_{i=1}^q 𝔍^{q−i}(𝔍 M_{p+i} + M_{p+i+1}) ⊂ ∑_{i=1}^{q+1} 𝔍^{q−i+1} M_{p+i}

and one has thus proved $(A_{q, p})$ and terminated the proof of (19.5.5).

Remarks (19.5.6).

(i) The results of this number retain a meaning and are valid when, instead of an $A$ -module $M$ and elements $f_{i} \in A$ , one takes an object $M$ of an arbitrary abelian category, and $n$ endomorphisms $f_{i}$ of $M$ which commute pairwise; the proofs adapt without difficulty to this more general case. The separation hypothesis for an $A$ -module $N$ relative to the $J$ -preadic topology must here be replaced by the hypothesis that every sub-object of $N$ contained in all the $\sum_{i} f_{i}^{r} (N)$ ( $r$ an arbitrary integer) is necessarily zero. The same remark applies to the results of 19.7.

(ii) Suppose that $A$ is a graded ring with degrees $⩾ 0$ . If $M$ (resp. each $g r_{p} (M)$ ) is a graded $A$ -module with degrees bounded below and if the $f_{i}$ contain no homogeneous component of degree 0, the separation hypothesis of (0, 15.1.9) is ipso facto verified for $M$ (resp. each $g r_{p} (M)$ ), and one therefore sees that in this case one has the implication c) ⟹ a) in (19.5.1) (resp. (19.5.5)).

(iii) Recall that, without separation hypothesis, the implication c) ⟹ b) is no longer valid even for $n = 1$ (0, 15.1.12, (iii)).

19.6. Regular sequences relative to a quotient filtered module

(19.6.1).

Let $A$ be a ring, $f = (f_{1}, \dots, f_{n})$ a finite sequence of elements of $A$ , $J = f_{1} A + \dots + f_{n} A$ the ideal it generates, and consider an exact sequence of $A$ -modules

$0 \to R \to N \to M \to 0$

where one supposes that $N$ is endowed with a decreasing filtration $(N_{k})$ formed of sub- $A$ -modules, with $N_{0} = N$ . One sets $R_{k} = R \cap N_{k}$ (filtration induced by $(N_{k})$ ), $M_{k} = p (N_{k})$ (quotient filtration of $(N_{k})$ ), and one denotes by $g r_{∙} (N)$ , $g r_{∙} (R)$ and $g r_{∙} (M)$ the graded $A$ -modules associated with these three filtrations. One sets moreover, for every $k$ ,

                       N'_k = N_k + 𝔍 N_{k-1} + ⋯ + 𝔍^{k-1} N_1 + 𝔍^k N_0

so that $M_{k}^{'} = p (N_{k}^{'}) = M_{k} + J M_{k - 1} + \dots + J^{k - 1} M_{1} + J^{k} M_{0}$ ; one sets moreover $R_{k}^{'} = R \cap N_{k}^{'}$ (filtration induced by $(N_{k}^{'})$ ), and one denotes by $g r_{∙}^{'} (N)$ , $g r_{∙}^{'} (M)$ and $g r_{∙}^{'} (R)$ the graded $A$ -modules associated with these three filtrations; one thus has a commutative diagram of exact sequences

  (19.6.1.1)         0 ──→ gr_•(R) ──→ gr_•(N) ──→ gr_•(M) ──→ 0
                              │            │            │
                              ▼            ▼            ▼
                     0 ──→ gr'_•(R) ──→ gr'_•(N) ──→ gr'_•(M) ──→ 0

where the vertical arrows are the canonical homomorphisms from a graded module associated with a filtration into the graded module associated with a coarser filtration.

One will note that, if one sets

                       R''_k = R_k + 𝔍 R_{k-1} + ⋯ + 𝔍^{k-1} R_1 + 𝔍^k R_0

one evidently has $R_{k}^{''} \subset R_{k}^{'}$ , but the filtrations $(R_{k}^{'})$ and $(R_{k}^{''})$ are in general distinct; one will denote by $g r_{∙}^{''} (R)$ the graded $A$ -module associated with the filtration $(R_{k}^{''})$ .

(19.6.2).

One has defined in (0, 15.1.5.2) the graded surjective homomorphisms of degree 0

                       ψ_N : (gr_•(N) ⊗_A (A/𝔍))[T_1, …, T_n] → gr'_•(N)
                       ψ_M : (gr_•(M) ⊗_A (A/𝔍))[T_1, …, T_n] → gr'_•(M)
                       ψ_R : (gr_•(R) ⊗_A (A/𝔍))[T_1, …, T_n] → gr''_•(R)

of which the first two are deduced from the last two vertical arrows of (19.6.1.1).

As the filtration $(R_{k}^{'})$ is finer than $(R_{k}^{''})$ on $R$ , one has a canonical homomorphism $γ : g r_{∙}^{''} (R) \to g r_{∙}^{'} (R)$ ; we shall denote by $ψ_{R}^{'}$ the composite homomorphism $γ \circ ψ_{R}$ . It then follows at once from the definitions that one has a commutative diagram

  (19.6.2.1)         (gr_•(R) ⊗_A (A/𝔍))[T_1, …, T_n]  ──ψ_R──→  gr''_•(R)
                                  │                                  │
                                  │ j'                               │ gr''_•(j)
                                  ▼                                  ▼
                     (gr_•(N) ⊗_A (A/𝔍))[T_1, …, T_n]  ──ψ_N──→  gr'_•(N)
                                  │                                  │
                                  │ p'                               │ gr'_•(p)
                                  ▼                                  ▼
                     (gr_•(M) ⊗_A (A/𝔍))[T_1, …, T_n]  ──ψ_M──→  gr'_•(M)
                                  │                                  │
                                  ▼                                  ▼
                                  0                                  0

where the columns are exact.

Proposition (19.6.3).

With the preceding notation, suppose that the sequence $f$ is $g r_{∙} (N)$ -regular. Consider the following properties:

a) $f$ is $g r_{∙} (M)$ -regular.
b) $ψ_{M}$ is bijective.
c) $ψ_{R}$ is surjective (in other words, $g r_{∙}^{''} (R)$ is generated as a $g r_{∙}^{'} (A)$ -module by $I m (g r_{∙} (R) \to g r_{∙}^{''} (R))$ ).
d) $ψ_{R}^{'}$ is injective.
d′) The homomorphism $(ψ_{R}^{'})_{0} : g r_{∙} (R) \otimes_{A} (A / J) \to g r_{∙}^{'} (R)$ obtained by restricting $ψ_{R}^{'}$ to polynomials of degree 0 is injective.
e) $ψ_{R}$ is bijective.
f) $j^{'}$ is injective.
f′) The homomorphism $g r_{∙} (R) \otimes_{A} (A / J) \to g r_{∙} (N) \otimes_{A} (A / J)$ obtained by restricting $j^{'}$ to polynomials of degree 0 is injective.
g) The $J$ -preadic filtration of $g r_{∙} (R)$ is induced by that of $g r_{∙} (N)$ .
h) The filtrations $(R_{k}^{'})$ and $(R_{k}^{''})$ on $R$ are identical.

One then has the following implications:

                              a) ⇒ e) ⇒ b) ⇔ c) ⇔ h)
                                    ⇕     ⇕
                                       g) ⇒ d) ⇔ d′) ⇔ f) ⇔ f′)

Moreover, when every quotient module of a submodule of a $g r_{k} (M)$ is separated for the $J$ -preadic topology (which will be the case when $A$ is Noetherian, the $f_{i}$ belong to the radical of $A$ , and the $g r_{k} (M)$ are $A$ -modules of finite type), then the conditions a) through h) are all equivalent.

Note that by (19.5.5), from the hypothesis that $f$ is $g r_{∙} (N)$ -regular, it follows that $ψ_{N}$ is bijective; as $g r_{∙}^{'} (j)$ is injective, the equivalence of d) and f) follows from the diagram (19.6.2.1), as does that of d′) and f′); moreover, f) and f′) are trivially equivalent, which shows the equivalence of d), d′), f) and f′).

Since we already know that $ψ_{R}$ is surjective and $ψ_{N}$ bijective, the equivalence of b) and c) also follows from the diagram (19.6.2.1) by a particular case of the five lemma.

The relation $ψ_{R}^{'} = γ \circ ψ_{R}$ (19.6.2) and the fact that $ψ_{R}$ is surjective show that condition c) is equivalent to saying that $γ$ is surjective, which is equivalent in turn to condition h); one has thus proved the equivalence of b), c) and h).

It is trivial that e) is equivalent to the conjunction of c) and d), hence also of b) and d). Note now that a) implies b), and that b) implies a) under the separation hypothesis (19.5.5). On the other hand, the implications a) ⇒ g) ⇒ f), and the implication f) ⇒ a) under the separation hypothesis, follow from (19.5.3). One has thus proved that a) entails e) (equivalent to the conjunction of b) and f′)), and also that all the conditions are equivalent under the separation hypothesis. Q.E.D.

Corollary (19.6.4).

Let $K$ be an ideal of $A$ , $L = J + K$ . Suppose that the filtration $(N_{k})$ (hence also $(M_{k})$ ) is the $K$ -preadic filtration, so that the filtrations $(N_{k}^{'})$ and $(M_{k}^{'})$ are the $L$ -preadic filtrations. Consider $g r_{∙} (R)$ (resp. $g r_{∙}^{''} (R)$ ) as a graded module over $g r_{∙} (A)$ (resp. $g r_{∙}^{'} (A)$ ).

(i) Let $S$ be a part of $g r_{∙} (R)$ which generates $g r_{∙} (R)$ as a $g r_{∙} (A)$ -module. Then condition c) of (19.6.3) is equivalent to the following condition:
c′) The image of $S$ in $g r_{∙}^{''} (R)$ by $ψ_{R}$ generates $g r_{∙}^{''} (R)$ as a $g r_{∙}^{'} (A)$ -module.
(ii) Suppose that condition e) of (19.6.3) is satisfied, and moreover that the quotients of the $g r_{k} (R)$ are separated for the $J$ -preadic topology (which will be the case when $A$ is Noetherian, the $f_{i}$ are in the radical of $A$ , and the $g r_{k} (N)$ are $A$ -modules of finite type). Then, in order for a part $S$ of $g r_{∙} (R)$ formed of homogeneous elements to generate $g r_{∙} (R)$ as a $g r_{∙} (A)$ -module, it is necessary and sufficient that its image by $ψ_{R}$ generate $g r_{∙}^{''} (R)$ as a $g r_{∙}^{'} (A)$ -module.

(i) Consider the homomorphism of degree 0

                       ψ_A : (gr_•(A) ⊗_A (A/𝔍))[T_1, …, T_n] → gr'_•(A) = gr_•^𝔏(A)

(0, 15.1.5.2). The fact that this homomorphism is surjective entails that the sub- $g r_{∙}^{'} (A)$ -algebra of $g r_{∙}^{''} (R)$ generated by $S^{'} = ψ_{R} (S)$ is none other than $I m (ψ_{R})$ , whence the equivalence of c) and c′).

(ii) Since e) is satisfied, the same is true of c), and by virtue of (i), it remains to prove the sufficiency of the condition in the statement. Now, as $ψ_{R}$ is bijective, saying that $S^{'}$ generates $g r_{∙}^{''} (R)$ regarded as a $g r_{∙}^{'} (A)$ -module is equivalent to saying that $S$ generates $(g r_{∙} (R) \otimes_{A} (A / J)) [T_{1}, \dots, T_{n}]$ as a $g r_{∙} (A)$ -module, and by virtue of the surjectivity of $ψ_{A}$ , one may, in this statement, replace the ring $g r_{∙} (A)$ by $(g r_{∙} (A) \otimes_{A} (A / J)) [T_{1}, \dots, T_{n}]$ . It evidently amounts to the same to say that $S$ generates $g r_{∙} (R) \otimes_{A} (A / J)$ as a module over the ring $g r_{∙} (A) \otimes_{A} (A / J)$ , or equivalently as a $g r_{∙} (A)$ -module. Now, if $T$ is the graded sub- $g r_{∙} (A)$ -module of $g r_{∙} (R)$ generated by $S$ , and if one sets $T^{'} = g r_{∙} (R) / T$ , one has by hypothesis $T^{'} / J T^{'} = 0$ , that is $J T^{'} = T^{'}$ ; but $T^{'}$ is a graded module, and each $T_{p}^{'}$ is a quotient of $g r_{p} (R)$ , hence separated for the $J$ -preadic topology; the hypothesis $J T_{p}^{'} = T_{p}^{'}$ therefore entails $T_{p}^{'} = 0$ for every integer $p$ , hence $T^{'} = 0$ , which completes the proof of the corollary.

19.7. Hironaka's normal flatness criterion

Theorem (19.7.1) (Hironaka).

Let $A$ be a ring, $K$ an ideal of $A$ , $f = (f_{1}, \dots, f_{n})$ a sequence of elements of $A$ which is $(A / K)$ -regular, $J = f_{1} A + \dots + f_{n} A$ the ideal generated by $f$ , $L = J + K$ , $M$ an $A$ -module.

One has in this case $(g r_{J}^{∙} (M)) \otimes_{A} (A / K) = (g r_{J}^{∙} (M)) \otimes_{A} (A / L) = (g r_{L}^{∙} (M)) \otimes_{A} (A / L)$ , because $(J^{n} M / J^{n + 1} M) \otimes_{A} (A / K) = J^{n} M / (J^{n + 1} + K J^{n}) M = L^{n} M / (L^{n + 1} + K L^{n}) M$ , the last equality following from Bourbaki, Alg., chap. II, 3rd ed., §3, n° 7, cor. 2 of prop. 6.

Consider the following conditions:

a) $g r_{L}^{∙} (M)$ is a flat $(A / L)$ -module.
b) $g r_{J}^{∙} (M)$ is a flat $(A / K)$ -module, and the canonical homomorphism (0, 15.1.5.2)

                       ψ_M : ((gr_𝔍^•(M)) ⊗_A (A/𝔏))[T_1, …, T_n] → gr_𝔏^•(M)

is bijective.

c) $(g r_{J}^{∙} (M)) \otimes_{A} (A / L)$ is a flat $(A / L)$ -module, and the sequence $f$ is $(g r_{J}^{∙} (M))$ -regular.
d) $g r_{L}^{∙} (M)$ is a flat $(A / L)$ -module, and for every $p \in A s s_{A} (A / K)$ , $(g r_{K}^{∙} (M))_{p}$ is a flat $(A_{p} / K_{p})$ -module.

One then has the implications

                              a) ⇒ c) ⇒ b)
                                       ⇑
                                       d)

When every quotient of a submodule of a $g r_{J}^{k} (M)$ $(k \geq 0)$ is ideally separated for $J$ (Bourbaki, Alg. comm., chap. III, §5, n° 1), the conditions a), b) and c) are equivalent.

Finally, when $A$ is Noetherian, the $f_{i}$ belong to the radical of $A$ , the sequence $f$ is $g r_{K}^{∙} (A)$ -regular, and $M$ is an $A$ -module of finite type, the conditions a), b), c), d) are equivalent.

The implication a) ⇒ c) is immediate (0, 15.1.13). If c) is satisfied, it follows from (19.5.5) that $ψ_{M}$ is bijective; moreover, $g r_{J}^{∙} (M) \otimes_{A} (A / L) = g r_{J}^{∙} (M) \otimes_{A} (A / K)$ is a flat $(A / L)$ -module, hence so is $(g r_{J}^{∙} (M) \otimes_{A} (A / L)) [T_{1}, \dots, T_{n}]$ , and consequently so is $g r_{L}^{∙} (M)$ since $ψ_{M}$ is an $(A / L)$ -isomorphism; thus c) entails b). The fact that a) implies b) shows at once that a) also implies d).

When every quotient of a $g r_{J}^{k} (M)$ is ideally separated for $J$ , it follows from (19.5.5) that condition b) entails that $f$ is $(g r_{J}^{∙} (M))$ -regular; moreover, since $(g r_{J}^{∙} (M) \otimes_{A} (A / L)) [T_{1}, \dots, T_{n}]$ , isomorphic to $g r_{L}^{∙} (M)$ , is then a flat $(A / L)$ -module, the same is true of $g r_{J}^{∙} (M) \otimes_{A} (A / L)$ , which is a direct factor of it; this proves therefore that b) then entails c). On the other hand, under the same hypothesis, c) entails a) by virtue of (0, 15.1.21) (where one may replace the Noetherian hypothesis by the hypothesis that $M$ is ideally separated for $J$ , by virtue of (III, 10.2.1)).

It remains therefore to prove that when $A$ is Noetherian, $M$ of finite type and the $f_{i}$ belong to the radical of $A$ , d) entails a).

By hypothesis there exists a free $A$ -module of finite type $N$ and an exact sequence $0 \to R \to N \to M \to 0$ . It suffices to prove that d) entails b), since every quotient of a submodule of a $g r_{J}^{k} (M)$ is then ideally separated for $J$ , by virtue of the fact that $g r_{J}^{k} (M)$ is an $A$ -module of finite type, that $J$ is contained in the radical of $A$ , and that $A$ is Noetherian (Bourbaki, Alg. comm., chap. III, §5, n° 1); in other words, the question is to see that $ψ_{M}$ is bijective.

As by hypothesis the sequence $f$ is $g r_{K}^{∙} (A)$ -regular, it is also $g r_{K}^{∙} (N)$ -regular; one may therefore apply (19.6.3), for every quotient module of a submodule of a $g r_{K}^{k} (M)$ is then separated for the $J$ -preadic topology, since the $f_{i}$ belong to the radical of $A$ by hypothesis. The commutative diagram (19.6.2.1) is therefore written here

                                                          0
                                                          │
                                                          ▼
  (19.7.1.1)   (gr'_•(R, N) ⊗_{A/𝔎} (A/𝔏))[T_1, …, T_n] ──ψ_R──→ gr''_•(R, N)
                              │                                       │
                              ▼                                       ▼
               (gr'_•(N) ⊗_{A/𝔎} (A/𝔏))[T_1, …, T_n]  ──ψ_N──→  gr'_•(N)
                              │                                       │
                              ▼                                       ▼
               (gr'_•(M) ⊗_{A/𝔎} (A/𝔏))[T_1, …, T_n]  ──ψ_M──→  gr'_•(M)
                              │                                       │
                              ▼                                       ▼
                              0                                       0

where $g r_{∙}^{'} (R, N)$ (resp. $g r_{∙}^{''} (R, N)$ ) is the graded module associated with $R$ for the filtration of the $R \cap K^{k} N$ (resp. $R \cap L^{k} N$ ); recall that in this diagram the columns are exact and that $ψ_{N}$

is bijective. The question is, by virtue of (19.6.3), to prove that $ψ_{R}$ is surjective, which will be done in several steps.

(19.7.1.2).

Set $B = g r_{K}^{∙} (A) \otimes_{A} (A / L)$ , $P = g r_{K}^{∙} (N) \otimes_{A} (A / L)$ , $Q = g r_{∙}^{''} (R, N)$ ; since $ψ_{N}$ is bijective, $P [T_{1}, \dots, T_{n}]$ is identified with $g r_{∙}^{'} (N)$ , and $Q$ with a sub- $B [T_{1}, \dots, T_{n}]$ -module of $P [T_{1}, \dots, T_{n}]$ ; we shall first see that one has

  (19.7.1.3)                  Q = (Q ∩ P)[T_1, …, T_n].

For this, set $Z = Q / ((Q \cap P) [T_{1}, \dots, T_{n}])$ ; this is a sub- $B [T_{1}, \dots, T_{n}]$ -module of $P [T_{1}, \dots, T_{n}] / ((Q \cap P) [T_{1}, \dots, T_{n}]) = (P / (Q \cap P)) [T_{1}, \dots, T_{n}]$ . As an $(A / L)$ -module, it is therefore isomorphic to a submodule of a direct sum of $(A / L)$ -modules isomorphic to $P / (Q \cap P) = (P + Q) / Q$ . But $(P + Q) / Q$ is an $(A / L)$ -module isomorphic to a submodule of $P [T_{1}, \dots, T_{n}] / Q$ , which is none other than $g r_{∙}^{'} (M)$ by virtue of the diagram (19.7.1.1). One has therefore

$(19.7.1.4) A s s_{A} (Z) \subset A s s_{A} (g r_{∙}^{'} (M)) .$

But each of the $g r_{L}^{k} (M)$ is, by hypothesis, a flat $(A / L)$ -module of finite type, hence projective since $A / L$ is Noetherian, and one has consequently $A s s_{A} (g r_{L}^{∙} (M)) \subset A s s_{A} (A / L)$ . To see that $Z = 0$ , it suffices therefore (Bourbaki, Alg. comm., chap. IV, §1, n° 1, cor. 1 of prop. 2 and n° 3, prop. 7) to see that for every $q \in A s s_{A / K} (A / L)$ , one has $Z_{q} = 0$ . But the ideals of $A s s_{A / K} (A / L)$ are the ideals of the form $p / (K / K)$ , where $p \in A s s_{A} (A / L)$ , and it amounts therefore to the same to see that $Z_{p} = 0$ for every $p \in A s s_{A} (A / L)$ . Now, if $p = r / K$ , where $r$ is a prime ideal of $A$ , one has $(g r_{J}^{∙} (M))_{p} = g r_{J}^{∙} (M_{r})$ and $(A / K)_{p} = A_{r} / K_{r}$ , and the images of the $f_{i}$ in $A_{r}$ form a $g r_{K}^{∙} (A_{r})$ -regular sequence (0, 15.1.14); applying to $A_{r}$ , $K_{r}$ and $M_{r}$ the implication a) ⇒ b) of the statement, one sees that the hypothesis d) entails that $ψ_{M}$ is bijective, hence also $ψ_{R}$ by virtue of (19.6.3); in other words, $Q_{p} = Q_{p}^{'} [T_{1}, \dots, T_{n}]$ , where one has set $Q^{'} = g r_{∙}^{'} (R, N) \otimes_{A} (A / L)$ , which is here a sub- $B$ -module of $P$ ; in particular $Q_{p}^{'} = Q_{p} \cap P_{p}$ , hence finally one has indeed $Z_{r} = Z_{p} = 0$ , which completes the proof of (19.7.1.3).

(19.7.1.5).

By virtue of (19.7.1.3), to prove that $ψ_{R}$ is surjective, it suffices to show that every homogeneous element of degree $m$ of $Q \cap P$ is the image of an element of the same degree of $g r_{∙}^{'} (R, N) \otimes_{A} (A / L)$ . Now, the elements of degree $m$ of $P$ are identified (by $ψ_{N}$ ) with the elements of $g r_{K}^{m} (N)$ belonging to $(K^{m} N + L^{m + 1} N) / L^{m + 1} N$ ; those which are images of elements of $g r_{∙}^{'} (R, N) \otimes_{A} (A / L)$ are identified with the elements of $g r_{K}^{m} (N)$ belonging to $((R \cap K^{m} N) + L^{m + 1} N) / L^{m + 1} N$ . It will therefore suffice in the end to prove, for every integer $m > 0$ , the inclusion

                  ((R ∩ 𝔏^m N) + 𝔏^{m+1} N) ∩ (𝔎^m N + 𝔏^{m+1} N) ⊂ (R ∩ 𝔎^m N) + 𝔏^{m+1} N.

As $K \subset L$ , one verifies at once that the first member of this relation is equal to $(R \cap (K^{m} N + L^{m + 1} N)) + L^{m + 1} N$ , so that everything reduces to proving

  (19.7.1.6)              R ∩ (𝔎^m N + 𝔏^{m+1} N) ⊂ (R ∩ 𝔎^m N) + 𝔏^{m+1} N.

We shall proceed by induction on $m$ , supposing therefore (19.7.1.6) satisfied when one replaces $m$ by an integer $m^{'} < m$ . We shall consider on the other hand, for $m$ fixed and an integer $d \geq 0$ , the relation

  (*_d)                  R ∩ (𝔎^m N + 𝔏^{m+1} N) ⊂ (R ∩ 𝔎^m N ∩ 𝔏^d N) + 𝔏^{m+1} N.

It is clear that it suffices to prove $(*_{d})$ for $d = m$ , and on the other hand $(*_{0})$ is trivially true. We shall prove $(*_{d})$ by induction on $d$ ; in other words, we suppose, for a $d \geq 0$ fixed (with $d < m$ ), that $(*_{d})$ is true, and we want to prove

  (*_{d+1})              R ∩ (𝔎^m N + 𝔏^{m+1} N) ⊂ (R ∩ 𝔎^m N ∩ 𝔏^{d+1} N) + 𝔏^{m+1} N.

Consider for this, for an $h \geq 0$ , the relation

  (**_{d+1, h})  R ∩ (𝔎^m N + 𝔏^{m+1} N) ⊂ (R ∩ (𝔏^{d+1} N + 𝔍^h 𝔎^d N) ∩ 𝔎^m N) + 𝔏^{m+1} N.

The hypothesis $(*_{d})$ implies that $(* *_{d + 1, 0})$ is true. We shall prove by induction on $h$ that $(* *_{d + 1, h})$ is true for every $h \geq 0$ . But one has the

Lemma (19.7.1.7).

Let $A$ be a Noetherian ring, $N$ an $A$ -module of finite type, $E$ , $F$ two submodules of $N$ , $L$ an ideal of $A$ . For every $k > 0$ there exists $h > 0$ such that

                       E ∩ (F + 𝔏^h N) ⊂ E ∩ F + 𝔏^k N.

Indeed, let $ϕ : N \to N / (E \cap F) = N_{1}$ be the canonical homomorphism, and set $E_{1} = ϕ (E)$ , $F_{1} = ϕ (F)$ , so that $E_{1} \cap F_{1} = 0$ and $ϕ (E \cap (F + L^{h} N)) = E_{1} \cap (F_{1} + L^{h} N_{1})$ . One knows $(0_{I}, 7.3.2)$ that there exists $h$ such that $(E_{1} + F_{1}) \cap L^{h} N_{1} \subset L^{k} (E_{1} + F_{1})$ ; if $h$ is so chosen and if $x_{1} \in E_{1}$ is such that there exists $y_{1} \in F_{1}$ for which $x_{1} - y_{1} \in L^{h} N_{1}$ , one deduces from the foregoing that there exist $u_{1} \in L^{k} E_{1}$ and $v_{1} \in L^{k} F_{1}$ such that $x_{1} = u_{1}$ and $y_{1} = v_{1}$ ; hence $E_{1} \cap (F_{1} + L^{h} N_{1}) \subset L^{k} N_{1}$ , which proves the lemma.

It suffices then to apply this lemma with $E = R \cap K^{m} N$ , $F = L^{d + 1} N$ and to take $k = m + 1$ to deduce that, with the corresponding value of $h$ , $(* *_{d + 1, h})$ entails $(*_{d + 1})$ .

We suppose therefore in all that follows that $(* *_{d + 1, h})$ is satisfied.

(19.7.1.8). Proof of $(* *_{d + 1, h + 1})$ : First case: $h \leq m - d$ .

Start therefore from an element

  (19.7.1.9)              g ∈ R ∩ (𝔏^{d+1} N + 𝔍^h 𝔎^d N) ∩ 𝔎^m N

congruent mod. $L^{m + 1} N$ to an element $g_{1} \in R \cap (K^{m} N + L^{m + 1} N)$ ; one has therefore

  (19.7.1.10)             g ∈ 𝔎^m N + 𝔏^{m+1} N.

It will suffice to prove that $g$ is also congruent mod. $L^{m + 1} N$ to an element $g_{2} \in R \cap (L^{d + 1} N + J^{h + 1} K^{d} N)$ , for that will entail $g_{2} \in K^{m} N$ , since one will have $g_{2} \in K^{d} N$ and $g_{2} - g \in L^{m + 1} N$ . In the case $h \leq m - d$ which we are considering, it suffices to demonstrate the relation

  (19.7.1.11)            (𝔏^{d+1} N + 𝔍^h 𝔎^d N) ∩ 𝔎^m N ⊂ 𝔍^{m-d+1} 𝔎^d N + 𝔏^{m+1} N

for the relations (19.7.1.9) and (19.7.1.10) entail that $g$ belongs to the first member of (19.7.1.11), and as $h \leq m - d$ , one has $J^{m - d + 1} \subset J^{h + 1}$ , which will establish $(* *_{d + 1, h + 1})$ . Moreover one has $K^{d} N \subset L^{d} N$ , and the relation

  (19.7.1.12)            𝔍^a N ∩ 𝔏^b N ⊂ 𝔏^{a+b-d} N

will entail (19.7.1.11). As $N$ is a free $A$ -module of finite type, one sees that it therefore suffices to prove

  (19.7.1.13)            𝔏^{m+1} ∩ 𝔎^d ⊂ 𝔍^{m-d+1} 𝔎^d.

It will suffice to prove that, in general, for $d < m$ , one has

  (19.7.1.14)            𝔏^{m+1} ∩ 𝔎^d ⊂ 𝔍^{m-d+1} 𝔎^d + 𝔎^{d+1}.

Indeed, as $K \subset L$ , hence $J^{m - d + 1} K^{d} \subset L^{m + 1}$ , the preceding relation will entail

                       𝔏^{m+1} ∩ 𝔎^{d+1} ⊂ 𝔍^{m-d} 𝔎^{d+1} + 𝔎^{d+2}

and by induction on $d \leq m$ , one will conclude

                       𝔏^{m+1} ∩ 𝔎^d ⊂ 𝔍^{m-d+1} 𝔎^d + 𝔎^{m+1}

whence (19.7.1.13), since $K^{m + 1} \subset J^{m - d + 1} K^{m - d + 1}$ .

As $L = J + K$ , the relation (19.7.1.14) is also written

  (19.7.1.15)        (𝔍^{m+1} + 𝔍^m 𝔎 + ⋯ + 𝔍 𝔎^m + 𝔎^{m+1}) ∩ 𝔎^d ⊂
                              𝔍^{m-d+1} 𝔎^d + 𝔍^{m-d} 𝔎^{d+1} + ⋯ + 𝔍 𝔎^m + 𝔎^{m+1} = 𝔍^{m-d+1} 𝔎^d + 𝔎^{d+1}.

We shall see that this inclusion is itself a consequence of

  (19.7.1.16)            𝔍^a 𝔎^b ∩ 𝔎^{a+1} ⊂ 𝔍^{a+1} 𝔎^{b-1}

valid for $a \geq 0$ , $b \geq 1$ . Indeed, to prove (19.7.1.15) it suffices to show that for $0 < q \leq d$ , one has

  (19.7.1.17)        (𝔍^{m+1} + 𝔍^m 𝔎 + ⋯ + 𝔍^{m-q+1} 𝔎^{q-1}) ∩ 𝔎^d ⊂
                              𝔍^{m+1-q} 𝔎^{q-1} 𝔎^{d-q+1} + ⋯ + 𝔎^{m+1} ⊂ 𝔍^{m+1-q} 𝔎^d

for, taking $q = d$ , this will entail (19.7.1.15). Now, to prove (19.7.1.17), it suffices to proceed by induction on $q$ supposing the relation true for $q \leq d$ ; an element of the first member is written, by hypothesis, as $y + z$ with $y \in J^{m + 1 - q} K^{q}$ , $z \in J^{m - q} K^{q + 1} + \dots + K^{m + 1}$ ; as $y + z \in K^{d} \subset K^{q + 1}$ , one deduces that $y \in K^{q + 1}$ , since $z \in K^{q + 1}$ ; taking account of (19.7.1.16), one has $y \in J^{m - q} K^{q + 1}$ , hence

                       y + z ∈ 𝔍^{m-q} 𝔎^{q+1} + ⋯ + 𝔍 𝔎^m + 𝔎^{m+1}, and y + z ∈ 𝔎^d

by hypothesis, which proves (19.7.1.17) where $q$ has been replaced by $q + 1$ .

It remains then to establish (19.7.1.16). An element of the first member is written $a_{1} f_{1} + \dots + a_{n} f_{n}$ , where the $a_{i}$ belong to $J^{a - 1} K^{b}$ ; taking account of the fact that the sequence $f$ is $(A / K^{b + 1})$ -regular by hypothesis, hence also $(A / K^{b + 1})$ -quasi-regular (0, 15.1.9), it follows from the definition (0, 15.1.7) that the relation $a_{1} f_{1} + \dots + a_{n} f_{n} \in K^{a + 1}$ , where the $a_{i} \in J^{a}$ , entails necessarily $a_{i} \in K^{a + 1}$ , hence $a_{i} \in J^{a - 1} K^{b} \cap K^{a + 1}$ ; it suffices to reason by

induction on $b$ for $b \geq 1$ (the case $b = 1$ being trivial) to complete the proof of (19.7.1.16) and consequently also of $(* *_{d + 1, h + 1})$ for $h \leq m - d$ .

(19.7.1.18). Proof of $(* *_{d + 1, h + 1})$ : Second case: $h > m - d$ .

We shall first demonstrate that, for every $h \geq 0$ , one has

  (19.7.1.19)        R ∩ (𝔏^{d+1} N + 𝔍^h 𝔎^d N) ⊂ 𝔍^h (R ∩ (𝔎^{d+1} N + 𝔍 𝔎^d N)) + 𝔏^{d+h+1} N.

Consider for this an element $g$ of the first member of (19.7.1.19). Note that one has $L^{d + 1} N + J^{h} K^{d} N = J^{d + 1} N + K^{d + 1} N$ since $L = K + J$ ; we shall consider the class ẑ of $g$ in $g r_{J}^{h} (g r_{K}^{d} (N))$ , which, by virtue of (19.5.5) and the hypothesis on $f$ , is identified with a sub- $(A / L)$ -module of $g r_{J}^{h} (N)$ . From the fact that $g \in R$ , we shall show that one has even (with the foregoing identification)

$(19.7.1.20) \overset{z}{^} \in g r_{J}^{h} (R, N) .$

Indeed, one has noted, in the proof of (19.7.1.3), that for every prime ideal $r$ of $A$ such that $p = r / K \in A s s_{A} (A / L)$ , the map $ψ_{R}$ is bijective, and consequently the relation (19.7.1.6), where one replaces all the modules by their localizations at $r$ , is true. As $g$ belongs to $R \cap (K^{m} N + L^{m + 1} N)$ , its image $g /1$ in $R_{r}$ belongs to $(R_{r} \cap K^{m} N_{r}) + L^{m + 1} N_{r}$ , hence the image $\overset{z}{^} /1$ of ẑ in $g r_{J}^{h} (N_{r})$ belongs to $g r_{J}^{h} (R_{r}, N_{r}) = (g r_{∙}^{'} (R, N))_{r} = (g r_{∙}^{''} (R, N))_{r}$ , where $q = r / L$ . In other words the image of ẑ by the canonical map $(g r_{J}^{h} (N))_{q} \to (g r_{J}^{h} (M))_{q}$ is zero for every $q \in A s s_{A / L} (A / L)$ . As one has seen in the proof of (19.7.1.3) that $A s s_{A / L} (g r_{J}^{h} (M)) \subset A s s_{A} (A / L)$ , one deduces (Bourbaki, Alg. comm., chap. IV, §1, n° 1, cor. 1 of prop. 2 and n° 3, prop. 7) that the image of ẑ by the canonical map $g r_{J}^{h} (N) \to g r_{J}^{h} (M)$ is zero, that is, one has the relation (19.7.1.20).

This being so, one may write by definition

  (19.7.1.21)            ẑ = Σ_{|p| = h} c_p f^p mod. 𝔏^{h+1} N

where one sets as usual $p = (p_{1}, \dots, p_{n})$ , $f^{p} = f_{1}^{p_{1}} f_{2}^{p_{2}} \dots f_{n}^{p_{n}}$ , $∣ p ∣ = p_{1} + \dots + p_{n}$ , and where $c_{p} \in K^{d} N$ . Moreover, since $ψ_{N}$ identifies $g r_{∙}^{'} (N)$ with $(g r_{K}^{∙} (N) \otimes_{A} (A / L)) [T_{1}, \dots, T_{n}]$ , the $c_{p}$ are determined mod. $K^{d + 1} N + L N$ , and if $\overset{c}{ˉ}_{p}$ is the class of $c_{p}$ mod. $K^{d + 1} N + L N$ , one has, by the preceding identification,

  (19.7.1.22)            ẑ = Σ c̄_p f^p.

Using (19.7.1.3), one deduces that one has necessarily, for every $p$ such that $∣ p ∣ = h$ ( $\overset{c}{ˉ}_{p}$ being this time identified by $ψ_{R}$ with an element of $g r_{K}^{d} (N)$ )

$(19.7.1.23) \overset{c}{ˉ}_{p} \in g r_{K}^{d} (R, N) .$

But since $d < m$ , the hypothesis that $(*_{d})$ is true implies that $\overset{c}{ˉ}_{p}$ belongs to the image of $ψ_{R}$ , hence one may suppose that $c_{p} \in (R \cap K^{d} N) + K^{d + 1} N + L N$ . The relation (19.7.1.21) then gives

                       ẑ ∈ 𝔍^h (R ∩ 𝔎^d N) + 𝔍^h 𝔎^{d+1} N + 𝔏^{h+1} N

and since $J \subset L$ , this proves (19.7.1.19).

In particular, one may therefore write

$g = t + g_{2}$

with $t \in J^{h} (R \cap K^{d} N)$ and $g_{2} \in K^{d + 1} N + L^{d + h + 1} N$ ; moreover, as $t \in R$ and $g \in R$ , one has $g_{2} \in R$ . On the other hand, using the hypothesis $h \geq m - d + 1$ , one sees that $t \in L^{m + 1} N$ . The element $g_{2}$ therefore answers to all the conditions stated at the beginning of (19.7.1.8), which completes the proof of (19.7.1).

Remarks (19.7.2).

(i) When one supposes that $g r_{K}^{∙} (A)$ is a flat $(A / K)$ -module, the hypothesis that the sequence $f$ is $(A / K)$ -regular entails that it is also $g r_{K}^{∙} (A)$ -regular (0, 15.1.14). One will note that this will be the case when $A$ and $A / K$ are regular rings (0, 17.3.6): in fact, $K$ is then a regular ideal of $A$ (19.1.2), hence quasi-regular, and by localizing at the maximal ideals of $A$ containing $K$ and using (0, 15.1.7), one sees that $g r_{K}^{∙} (A)$ is a projective $(A / K)$ -module.

(ii) It would be interesting to be able to prove the last conclusion of (19.7.1) by supposing only that there exists an $A$ -algebra $B$ which is a Noetherian ring for which $L B$ is contained in the radical of $B$ , and that $M$ is a $B$ -module of finite type.

Corollary (19.7.3).

Let $A$ be a Noetherian local ring, $m$ its maximal ideal, $K$ an ideal of $A$ such that the ring $A / K$ is regular and of dimension $n$ , $M$ an $A$ -module of finite type. The following conditions are equivalent:

a) $g r_{K}^{∙} (M)$ is a flat $(A / K)$ -module.
b) If $P (T)$ and $Q (T)$ are the Poincaré series of the $(A / m)$ -graded modules $g r_{m}^{∙} (M)$ and $g r_{K}^{∙} (M) \otimes_{A} (A / m)$ , one has

$(19.7.3.1) Q (T) = P (T) (1 - T)^{n} .$

Let $f = (f_{1}, \dots, f_{n})$ be a sequence of elements of $A$ such that the images of the $f_{i}$ in $A / K$ form a regular system of parameters of $A / K$ (0, 17.1.6), so that $f$ is an $(A / K)$ -regular sequence; moreover, if $J$ is the ideal generated by $f$ , one has $J + K = m$ , and as $A / m$ is a field, $g r_{K}^{∙} (M)$ is a flat $(A / m)$ -module. As $A$ is Noetherian and $M$ an $A$ -module of finite type, one may apply the equivalence of a) and b) in (19.7.1). Moreover, as $A / m$ is a field, and $ψ_{M}$ is surjective, the fact that $ψ_{M}$ is bijective is equivalent to saying that for every $h \geq 0$ , $g r_{m}^{h} (M)$ and the submodule of elements of degree $h$ of $(g r_{K}^{∙} (M) \otimes_{A} (A / m)) [T_{1}, \dots, T_{n}]$ have the same rank over $A / m$ . But for the second of these modules, the rank in question is evidently equal to

                       Σ_{i+j = h} binom(n + j − 1, j) rank_{A/𝔪}(gr_𝔎^i(M) ⊗_A (A/𝔪)).

As $(1 - T)^{- n} = Σ_{j} bin o m (n + j - 1, j) T^{j}$ , this proves that the relation (19.7.3.1) is necessary and sufficient for $ψ_{M}$ to be bijective.

Corollary (19.7.4).

Let $X$ be a locally Noetherian prescheme, $Y$ a regular and connected closed subprescheme of $X$ , $F$ a coherent $O_{X}$ -Module normally flat (6.10.1) along $Y$ . Then there exists a formal series $R \in Q [[T]]$ (independent of $x \in Y$ ) such that, for every $x \in Y$ , the Poincaré series of $g r_{m}^{∙} (F_{x})$ is given by the formula

  (19.7.4.1)             P_{ℱ(x)}(T) = R(T)(1 − T)^{-n}    where n = dim(𝒪_x).

Let $J$ be the coherent Ideal of $O_{X}$ defining $Y$ ; the hypothesis on $F$ signifies that $g r_{J}^{∙} (F)$ is a flat $O_{Y}$ -Module. One may therefore, for every $x \in Y$ , apply (19.7.3) taking $A = O_{X, x}$ , $K = J_{x}$ , $M = F_{x}$ ; the hypothesis a) being fulfilled by hypothesis, the Poincaré series $P_{F (x)} (T)$ is therefore equal to $R_{x} (T) (1 - T)^{- n}$ , where $R_{x} (T)$ is the Poincaré series of $g r_{J}^{∙} (F) \otimes_{O_{Y}} k (x)$ . Now, each of the $O_{Y}$ -Modules $J^{k} F / J^{k + 1} F$ is flat and coherent by hypothesis, hence locally free (2.1.12), and since $Y$ is supposed connected, each $J^{k} F / J^{k + 1} F$ is of constant rank on $Y$ , which proves that the Poincaré series $R_{x} (T)$ is independent of $x \in Y$ .

Corollary (19.7.5).

Let $X$ be a locally Noetherian prescheme, $Y$ a closed subprescheme of $X$ , $Z$ a closed subprescheme of $Y$ ; one supposes $Y$ and $Z$ regular. Let $F$ be a coherent $O_{X}$ -Module.

(i) If $F$ is normally flat along $Y$ at the points of $Z$ (cf. (11.3.4) for the definition of normal flatness at a point), then $F$ is normally flat along $Z$ .

(ii) Suppose moreover that $Z$ is connected and that $O_{X}$ is normally flat along $Y$ at the points of $Z$ (which will hold in particular if $X$ is regular at the points of $Z$ ). If $F$ is normally flat along $Z$ and if there exists a point $z \in Z$ such that $F$ is normally flat along $Y$ at the point $z$ , then $F$ is normally flat along $Y$ at the points of $Z$ .

Let $J_{Y}$ and $J_{Z}$ be the coherent Ideals of $O_{X}$ defining respectively $Y$ and $Z$ . For every point $z \in Z$ , the canonical immersion $Z \to Y$ is regular at the point $z$ , since $Y$ and $Z$ are regular at that point (19.1.1); in other words, there exists a sequence $(f_{i})_{1 \leq i \leq n}$ of elements of $J_{Y, z}$ which is $O_{Z, z}$ -regular and is such that $J_{Z, z} = J_{Y, z} + Σ f_{i} O_{X, z}$ .

(i) The hypothesis signifies that $g r_{J_{Y}}^{∙} (F_{z})$ is a flat $O_{Y, z}$ -module; it therefore entails that $g r_{J_{Z}}^{∙} (F_{z})$ is a flat $O_{Z, z}$ -module, by virtue of the fact that in (19.7.1), a) entails b).

(ii) The supplementary hypothesis on $X$ entails that the sequence $(f_{i})$ is $g r_{J_{Y}}^{∙} (F_{z})$ -regular at every point $z \in Z$ (19.7.2). On the other hand, $Z$ being regular and connected, is integral, and if $g r_{J_{Y}}^{∙} (F_{z})$ is $O_{Y, z}$ -flat at a point $z \in Z$ , then, as the generic point $ξ$ of $Z$ is a generization of $z$ , one concludes that $g r_{J_{Y}}^{∙} (F_{ξ})$ is $O_{Y, ξ}$ -flat $(0_{I}, 6.3.1)$ . One deduces that if at every point $z^{'}$ of $Z$ , $g r_{J_{Y}}^{∙} (F_{z^{'}})$ is a flat $O_{Y, z^{'}}$ -module, then $g r_{J_{Y}}^{∙} (F_{z})$ is a flat $O_{Y, z}$ -module, for one may apply the equivalence of a) and d) in (19.7.1).

19.8. Properties of passage to projective limit

In this number, the notation and conventions on projective limits are those of (8.5.1) and (8.8.1).

Proposition (19.8.1).

Suppose that the transition morphisms $S_{μ} \to S_{λ}$ ( $λ \leq μ$ ) are flat, and moreover that one of the two following hypotheses is satisfied:

1° The preschemes $S_{λ}$ are locally Noetherian.
2° The transition morphisms $S_{μ} \to S_{λ}$ are surjective (hence faithfully flat).

Under these conditions:

(i) Suppose $S_{α}$ quasi-compact. Let $F_{α}$ be a quasi-coherent $O_{S_{α}}$ -Module, which is moreover supposed of finite type when hypothesis 1° is satisfied. Let $(f_{i, α})_{1 \leq i \leq n}$ be a finite sequence of sections of $O_{S_{α}}$ above $S_{α}$ . In order for the sequence of sections $f_{i}$ of the $O_{S}$ -Module $F$ above $S$ , corresponding to the $f_{i, α}$ ( $1 \leq i \leq n$ ), to be $F$ -regular, it is necessary and sufficient that there exist $λ \geq α$ such that the sequence of sections $f_{i, λ}$ of $F_{λ}$ above $S_{λ}$ be $F_{λ}$ -regular.

(ii) Suppose $X_{α}$ quasi-compact, and let $j_{α} : X_{α} \to S_{α}$ be an immersion, which one supposes locally of finite presentation when hypothesis 2° is satisfied. Then, in order for the corresponding immersion $j : X \to S$ to be regular, it is necessary and sufficient that there exist $λ \geq α$ such that $j_{λ} : X_{λ} \to S_{λ}$ be regular.

(i) If $p_{α} : S \to S_{α}$ is the canonical projection, one has $F / (Σ f_{i} F) = p_{α}^{*} (F_{α} / (Σ f_{i, α} F_{α}))$ , hence one is reduced to the case $n = 1$ , in which case one suppresses the index $i$ . The fact that the condition is sufficient follows from the fact that $p_{α}$ is flat (8.3.8) and from (0, 15.1.5). In case 2°, $p_{α}$ is faithfully flat (8.3.8) and the necessity of the condition still follows from (0, 15.2.5), with $λ = α$ . In case 1°, denote by $N_{α}$ (resp. $N$ ) the kernel of the homomorphism $F_{α} \to F_{α}$ (resp. $F \to F$ ), multiplication by $f_{α}$ (resp. by $f$ ); since $F_{α}$ is coherent by hypothesis, so is $N_{α}$ , hence the hypothesis $N = 0$ entails $N_{λ} = 0$ for a $λ \geq α$ by virtue of (8.5.8, (ii)).

(ii) Since $p_{α}$ is flat, the sufficiency of the condition follows from (19.1.5, (ii)). To prove its necessity, note that, since $j_{α} (X_{α})$ is quasi-compact, it is contained in a quasi-compact open of $S_{α}$ and one may therefore limit oneself to the case where $S_{α}$ is also quasi-compact and $j_{α}$ a closed immersion, so that the image of $X_{α}$ is defined by a quasi-coherent Ideal $J_{α}$ of $O_{S_{α}}$ , which one may moreover suppose of finite type in cases 1° and 2° ( $j_{α}$ being locally of finite presentation in case 2°); the image of $X_{α}$ (resp. $X$ ) in $S_{α}$ (resp. $S$ ) is then defined by $J_{λ} = (J_{α})_{λ}$ (resp. $J = (J_{α})$ ) which is still of finite type. One may moreover, taking account of (8.2.11), suppose that $J$ is generated by a regular sequence $(f_{i})_{1 \leq i \leq n}$ of sections above $S$ , which therefore define a surjective homomorphism $u : O_{S}^{n} \to J$ . Taking account of (8.5.2, (i)) and (8.5.7), there exist $λ \geq α$ and a surjective homomorphism $u_{λ} : O_{S_{λ}}^{n} \to J_{λ}$ such that $u = p_{λ}^{*} (u_{λ})$ , hence the $f_{i}$ are the canonical images of sections $f_{i, λ}$ of $J_{λ}$ above $S_{λ}$ generating that Ideal. By virtue of (i), there exists therefore $μ \geq λ$ such that the sequence $(f_{i, μ})$ is $O_{S_{μ}}$ -regular, hence the immersion $j_{μ}$ is regular.

Proposition (19.8.2).

Let $X_{α}$ be a quasi-compact $S_{α}$ -prescheme, locally of finite presentation.

(i) Let $F_{α}$ be a quasi-coherent $O_{X_{α}}$ -Module of finite presentation, $(f_{i, α})_{1 \leq i \leq n}$ a sequence of sections of $O_{X_{α}}$ above $X_{α}$ . In order for the sequence of sections $f_{i}$ of $O_{X}$ above $X$ , corresponding to the $f_{i, α}$ , to be $F$ -transversally regular relative to $S$ (19.2.1), it is necessary and sufficient that there exist $λ \geq α$ such that the sequence of sections $f_{i, λ}$ of $O_{X_{λ}}$ above $X_{λ}$ be $F_{λ}$ -transversally regular relative to $S_{λ}$ .

*(ii) Let $Y_{α}$ be a quasi-compact $S_{α}$ -prescheme, $j_{α} : Y_{α} \to X_{α}$ an $S_{α}$ -immersion locally of finite

presentation. In order for the corresponding $S$ -immersion $j : Y \to X$ to be transversally regular relative to $S$ , it is necessary and sufficient that there exist $λ \geq α$ such that $j_{λ} : Y_{λ} \to X_{λ}$ be transversally regular relative to $S_{λ}$ .*

(i) The sufficiency of the condition follows from (0, 15.1.15). To prove that it is necessary, it will suffice to show that every point $x \in X$ has an open neighbourhood $V (x)$ for which there exist an index $λ = λ (x) \geq α$ and an open neighbourhood $V_{λ}$ of the image $x_{λ}$ of $x$ in $X_{λ}$ such that $V (x)$ is the inverse image of $V_{λ}$ and that the sequence of the restrictions to $V_{λ}$ of the $f_{i, λ}$ is $(F_{λ} ∣ V_{λ})$ -transversally regular relative to $S_{λ}$ : indeed, it will then suffice to apply (8.3.4) to the open set $U_{λ}$ of $X_{λ}$ , defined by the condition of being the largest open such that the restrictions to $U_{λ}$ of the $f_{i, λ}$ form a sequence $(F_{λ} ∣ U_{λ})$ -transversally regular relative to $S_{λ}$ . Let $s$ be the image of $x$ in $S$ , and for every $λ \geq α$ , let $s_{λ}$ be the image of $s$ in $S_{λ}$ . Then the fibre $X_{s}$ is the projective limit of the fibres $(X_{λ})_{s_{λ}}$ and is locally Noetherian since the morphism $X \to S$ is locally of finite presentation; as moreover $(X_{α})_{s_{α}}$ is quasi-compact and the transition morphisms $(X_{μ})_{s_{μ}} \to (X_{λ})_{s_{λ}}$ are flat (since this is so of $Spec (k (s_{μ})) \to Spec (k (s_{λ}))$ ), one may apply to the sections $f_{i} \otimes 1$ of $F_{s} = F \otimes_{O} k (s)$ above $X_{s}$ the result of (19.8.1, (i)); whence the existence of $λ \geq α$ such that the sections $f_{i, λ} \otimes 1$ of $(F_{λ})_{s_{λ}} = F_{λ} \otimes_{O} k (s_{λ})$ above $(X_{λ})_{s_{λ}}$ form a $(F_{λ})_{s_{λ}}$ -regular sequence. Moreover (11.2.6), one may suppose that $F_{λ}$ is $S_{λ}$ -flat at the point $x_{λ}$ . It then follows from (11.3.8) that there exists an open neighbourhood $V_{λ}$ of $x_{λ}$ in $X_{λ}$ satisfying the required conditions.

(ii) The sufficiency of the condition follows from (19.2.7, (ii)). To prove its necessity, it again suffices to show that every point $x \in j (Y)$ admits an open neighbourhood $V (x)$ for which there exist an index $λ = λ (x) \geq α$ and an open neighbourhood $V_{λ}$ of the image $x_{λ}$ of $x$ in $X_{λ}$ such that $V (x)$ is the inverse image of $V_{λ}$ and that the restriction $j_{λ}^{- 1} (V_{λ}) \to V_{λ}$ of $j_{λ}$ is transversally regular relative to $S_{λ}$ . Let $s$ be the image of $x$ in $S$ , and for every $λ \geq α$ , let $s_{λ}$ be the image of $s$ in $S_{λ}$ . As $X_{s}$ (resp. $Y_{s}$ ) is the projective limit of the fibres $(X_{λ})_{s_{λ}}$ (resp. $(Y_{λ})_{s_{λ}}$ ), one may, as in (i), use (19.8.1, (ii)), and as by hypothesis the immersion $Y_{s} \to X_{s}$ is regular, there exists an index $λ (s)$ such that for $λ \geq λ (s)$ the immersion $(Y_{λ})_{s_{λ}} \to (X_{λ})_{s_{λ}}$ be regular. Moreover, by virtue of (19.2.4) there exists a quasi-compact open neighbourhood $W (x)$ of $x$ in $X$ such that the structure morphisms $W (x) \to S$ and $Y \cap W (x) \to S$ are flat; applying (11.2.6) and (8.2.11), one may suppose that there exist $λ \geq λ (s)$ and a quasi-compact open neighbourhood $W (x_{λ})$ of $x_{λ}$ in $X_{λ}$ , such that $W (x)$ is the inverse image of $W (x_{λ})$ and that the restrictions to $W (x_{λ})$ and to $Y_{λ} \cap W (x_{λ})$ respectively of the structure morphisms $X_{λ} \to S_{λ}$ and $Y_{λ} \to S_{λ}$ be flat. One concludes therefore from (19.2.4) that there exists an open neighbourhood $V_{λ}$ of $x_{λ}$ answering the question.

Remark (19.8.3).

The preceding proofs show that the statements of (19.8.1) and (19.8.2) subsist when one replaces the conditions concerning regularity or transversal regularity in all of $X$ (resp. $X_{λ}$ ) by these conditions in a neighbourhood of a point $x \in X$ (resp. in a neighbourhood of $x_{λ}$ , projection of $x$ ).

19.9. $F$ -regular sequences and depth

(19.9.1).

Recall that if $X$ is a locally Noetherian prescheme, $F$ a coherent $O_{X}$ -Module, $T$ a part of $X$ , one sets (5.10.1)

  (19.9.1.1)             prof_T(ℱ) = inf_{t ∈ T} prof(ℱ_t).

One sets moreover, for every point $t \in T$ ,

  (19.9.1.2)             prof_{T, t}(ℱ) = inf_{z ∈ T ∩ Spec(𝒪_{X, t})} prof(ℱ_z),

and one says that $p ro f_{T, t} (F)$ is the $T$ -depth of $F$ at the point $t$ ; it is clear that one has

  (19.9.1.3)             prof_T(ℱ) = inf_{t ∈ T} prof_{T, t}(ℱ).

Lemma (19.9.2).

Let $A$ be a Noetherian ring, $M$ an $A$ -module of finite type, $J$ an ideal of $A$ . In order that, for every prime ideal $p \supset J$ , one have $p ro f (M_{p}) \geq r$ , it is necessary and sufficient that there exist an $M$ -regular sequence of $r$ elements belonging to $J$ .

The sufficiency of the condition follows at once from (0, 16.4.5); let us prove its necessity.

We reason by induction on $r$ (there is nothing to prove if $r = 0$ ): since $p ro f (M_{p}) \geq r - 1$ for every $p \supset J$ , there exists by hypothesis an $M$ -regular sequence $(g_{i})_{1 \leq i \leq r - 1}$ formed of elements of $J$ . Set $N = M / (g_{1} M + \dots + g_{r - 1} M)$ ; for every $p \supset J$ , the images of the $g_{i}$ in $A_{p}$ therefore belong to the maximal ideal $p A_{p}$ , and form an $M_{p}$ -regular sequence (0, 15.1.14); one concludes therefore from (0, 16.4.6) that one has $p ro f (N_{p}) = p ro f (M_{p}) - (r - 1) \geq 1$ , and one sees that everything reduces to proving the lemma for $r = 1$ . Now the hypothesis then signifies that, for every $p \supset J$ , $p A_{p}$ is not associated to $M_{p}$ (0, 16.4.6), hence (Bourbaki, Alg. comm., chap. IV, §1, n° 2, prop. 5), $p$ is not associated to $M$ . In other words, $J$ is not contained in any of the prime ideals of $A ss (M)$ , hence it is not contained in their union (Bourbaki, Alg. comm., chap. II, §1, n° 1, prop. 2). But as this union is the set of elements which are not $M$ -regular (Bourbaki, Alg. comm., chap. IV, §1, n° 1, cor. 2 of prop. 2), this proves the lemma.

Proposition (19.9.3).

Let $X$ be a locally Noetherian prescheme, $F$ a coherent $O_{X}$ -Module, $Y$ a closed part of $X$ , $y$ a point of $Y$ , $n$ an integer > 0. The following conditions are equivalent:

a) There exists an open neighbourhood $U$ of $y$ in $X$ such that $p ro f_{Y \cap U} (F ∣ U) \geq n$ .
b) There exists an open neighbourhood $U$ of $y$ in $X$ and a sequence $(f_{i})_{1 \leq i \leq n}$ of $n$ sections of $O_{X}$ above $U$ , which is $(F ∣ U)$ -regular, such that $f_{i} (x) = 0$ $(0_{I}, 5.5.1)$ for $1 \leq i \leq n$ at every point $x \in U \cap Y$ .
c) One has $p ro f_{Y, y} (F) \geq n$ .

Moreover, the set of $y \in Y$ satisfying these conditions is open in $Y$ .

The last assertion follows trivially from a). The equivalence of a) and b) follows from (19.9.2) applied to an affine open neighbourhood of $y$ in $X$ and to (0, 15.1.14). It is clear that a) entails c), every open neighbourhood of $y$ in $X$ containing $Spec (O_{X, y})$ ; let us show conversely that c) implies b). It follows from c) and from the definition (19.9.1.1) that one may

apply (19.9.2) to the ring $O_{X, y}$ , to the $O_{X, y}$ -module $F_{y}$ and to the ideal $J_{y}$ of $O_{X, y}$ , hence there exists an $F_{y}$ -regular sequence $(g_{i})_{1 \leq i \leq n}$ of $n$ elements of $J_{y}$ . There exists therefore an open neighbourhood $U$ of $y$ in $X$ and a sequence $(f_{i})_{1 \leq i \leq n}$ of sections of $J$ above $U$ such that the $g_{i}$ are the germs of the $f_{i}$ at $y$ ; using (0, 15.2.4), one deduces that there exists an open neighbourhood $U^{'} \subset U$ of $y$ in $X$ such that the $f_{i} ∣ U^{'}$ form an $(F ∣ U^{'})$ -regular sequence.

Corollary (19.9.4).

With the notation of (19.9.3), the function $y \mapsto p ro f_{Y, y} (F)$ is lower semi-continuous in $Y$ .

Proposition (19.9.5).

With the notation of (19.9.3), let $X^{'}$ be a locally Noetherian prescheme, $g : X^{'} \to X$ a flat morphism, $Y^{'} = g^{- 1} (Y)$ , $F^{'} = g^{*} (F)$ . Then, for every point $y^{'} \in Y^{'}$ above $y$ , one has $p ro f_{Y^{'}, y^{'}} (F^{'}) = p ro f_{Y, y} (F)$ .

Indeed, for every $z^{'} \in Spec (O_{X^{'}, y^{'}})$ , it follows from $(0_{I}, 6.3.1)$ that one has $p ro f (F_{z^{'}}^{'}) \geq p ro f (F_{g (z^{'})})$ , and if z'' is a maximal point of the fibre $g^{- 1} (g (z^{'}))$ , generization of $z^{'}$ (hence belonging also to $Spec (O_{X^{'}, y^{'}})$ ), one has $(0_{I}, 6.3.1)$ $p ro f (F_{z^{''}}^{'}) = p ro f (F_{g (z^{''})})$ . The proposition follows from the fact that one has $g (Spec (O_{X^{'}, y^{'}})) = Spec (O_{X, g (y^{'})})$ since $g$ is flat (2.3.4).

Proposition (19.9.6).

Let $f : X \to S$ be a morphism locally of finite presentation, $Y$ a closed part of $X$ , $F$ an $O_{X}$ -Module of finite presentation and $f$ -flat, $n$ an integer > 0. For every point $y \in Y$ , the following conditions are equivalent (denoting by $X_{f (y)}$ the fibre of $f$ at the point $f (y)$ , by $F_{f (y)}$ the inverse image of $F$ in $X_{f (y)}$ , and setting $Y_{f (y)} = X_{f (y)} \cap Y$ ):

a) One has $p ro f_{Y_{f (y)}, y} (F_{f (y)}) \geq n$ .
b) There exist an open neighbourhood $U$ of $y$ in $X$ and a sequence $(g_{i})_{1 \leq i \leq n}$ of $n$ sections of $O_{X}$ above $U$ , which is transversally $F$ -regular relative to $S$ (cf. (19.2.1) for the terminology) and such that $g_{i} (x) = 0$ for $1 \leq i \leq n$ and for every $x \in U \cap Y$ .

The set $V_{n}$ of $y \in Y$ satisfying these conditions is open in $Y$ ; if moreover $Y$ is locally constructible in $X$ , $V_{n}$ is retrocompact in $X$ $(0_{III}, 9.1.1)$ .

By virtue of (19.9.3), condition a) is equivalent to the existence of an open neighbourhood $U$ of $y$ in $X$ and of an $(F_{f (y)} ∣ (U \cap X_{f (y)}))$ -regular sequence $(h_{i})_{1 \leq i \leq n}$ of sections of $O_{X}$ above $U \cap X_{f (y)}$ such that $h_{i} (x) = 0$ for every $x \in U \cap Y_{f (y)}$ . If one again denotes by $Y$ a closed subprescheme of $X$ having $Y$ as underlying space, by $J$ the quasi-coherent Ideal of $O_{X}$ defining $Y$ , one may again say that one has $h_{i} \in J_{f (y)}$ for $1 \leq i \leq n$ . As one may replace $U$ by a smaller affine neighbourhood of $y$ , one may suppose that the $h_{i}$ are the images of a sequence $(g_{i})_{1 \leq i \leq n}$ of sections of $J$ above $U$ , so that the germs $(g_{i})_{y}$ belong to the maximal ideal $m_{y}$ of $O_{X, y}$ . The equivalence of a) and b) then follows from (11.3.8), which also proves that the set $V_{n}$ is open in $Y$ . It remains to prove the last assertion; as it is again a question of local properties on $X$ , one may suppose $S = Spec (A)$ affine and $X$ of finite presentation over $S$ ; there exists then a Noetherian subring A_0 of $A$ , a prescheme X_0 of finite type over $S_{0} = Spec (A_{0})$ and a coherent $O_{X_{0}}$ -Module $F_{0}$ such that $X = X_{0} \times_{S_{0}} S$ and $F = F_{0} \otimes_{O_{X_{0}}} O_{X}$ (8.9.1); one may moreover suppose that, if $f_{0} : X_{0} \to S_{0}$ is the structure morphism, $F_{0}$ is $f_{0}$ -flat (11.2.6). Since $Y$ is constructible, one may moreover (8.3.11) suppose

that $Y = p^{- 1} (Y_{0})$ , where $p : X \to X_{0}$ is the canonical projection. Then, for every $y \in Y$ , it follows from (19.9.5) and the transitivity of fibres that one has

                       prof_{Y_{f(y)}, y}(ℱ_{f(y)}) = prof_{(Y_0)_{f_0(y_0)}, y_0}((ℱ_0)_{f_0(y_0)})

setting $y_{0} = p (y)$ ; if $V_{n}^{0}$ is the set of $y_{0} \in Y_{0}$ for which the second member of this relation is $\geq n$ , one has therefore $V_{n} = p^{- 1} (V_{n}^{0})$ . As X_0 is Noetherian, $V_{n}^{0}$ is a retrocompact open in X_0, and consequently (cf. proof of (1.8.2)) $V_{n}$ is retrocompact in $X$ .

Corollary (19.9.7).

Under the general hypotheses of (19.9.6), the function $y \mapsto p ro f_{Y_{f (y)}, y} (F_{f (y)})$ is lower semi-continuous in $Y$ . If moreover $Y$ is locally constructible, this function is locally constructible.

Proposition (19.9.8).

Let $f : X \to S$ be a morphism locally of finite presentation, $Z$ a closed part of $X$ , $F$ an $O_{X}$ -Module of finite presentation and $f$ -flat, $G$ a quasi-coherent $O_{S}$ -Module. Suppose that, for every $x \in Z$ , one has $p ro f ((F_{f (x)})_{x}) \geq 1$ (resp. $p ro f ((F_{f (x)})_{x}) \geq 2$ ). Then, if $j : X - Z \to X$ is the canonical injection, and if one sets $H = F \otimes_{O} f^{*} (G)$ , the canonical homomorphism $ρ_{G} : H \to j_{*} (j^{*} (H))$ relative to $j$ is injective (resp. bijective).

The question is evidently local on $X$ and $S$ , and it suffices to demonstrate the proposition in a neighbourhood of a point $x \in Z$ . In other words, one may restrict to the case where $X$ and $S$ are affine. Thanks to (19.9.6), one may suppose that there exists a transversally $F$ -regular section $g_{1}$ of $O_{X}$ above $X$ relative to $S$ (resp. a transversally $F$ -regular sequence $(g_{1}, g_{2})$ of two sections of $O_{X}$ above $X$ relative to $S$ ) such that, if $Z^{'}$ is the set of $x^{'} \in X$ for which $g_{1} (x^{'}) = 0$ (resp. $g_{1} (x^{'}) = g_{2} (x^{'}) = 0$ ), one has $Z \subset Z^{'}$ .

This being so, let us return to the proof of (19.9.8); by virtue of (19.9.6), one still has $p ro f ((F_{f (x)})_{x}) \geq 1$ (resp. $p ro f ((F_{f (x)})_{x}) \geq 2$ ) for every $x \in Z^{'}$ . If the proposition is proved replacing the pair $(X, Z)$ by (X, Z') and $(X - Z, Z^{'} \cap (X - Z))$ , it is immediate that it will also be so for $(X, Z)$ ; one may therefore restrict to proving it for $X$ and $Z^{'}$ . In other words, one may restrict to demonstrating (19.9.8) when $S = Spec (A)$ , $X = Spec (B)$ , where $B$ is an $A$ -algebra of finite presentation, $Z = V (J)$ , where $J$ is an ideal of finite type of the ring $B$ , $G = N$ and $F = M$ , where $N$ is an $A$ -module and $M$ a $B$ -module of finite presentation which is a flat $A$ -module. One may moreover reduce to the case where $N$ is an $A$ -module of finite presentation. Indeed, $N$ is the inductive limit of a filtered inductive system of $A$ -modules of finite presentation $N_{α}$ (by double inductive limit); if one sets $H_{α} = F \otimes_{O} f^{*} (N_{α})$ , $H$ is the inductive limit of the $H_{α} = F \otimes_{O} f^{*} (N_{α})$ ; as $j^{*}$ and $j_{*}$ commute with inductive limits and lim is an exact functor in the category of quasi-coherent Modules, if the proposition is proved for each of the $H_{α}$ , it will be so for $H$ .

It suffices moreover to prove that the canonical homomorphism

                       Γ(X, ℋ) → Γ(X − Z, ℋ)

is injective (resp. bijective). There exists then a Noetherian subring A_0 of $A$ , an A_0-algebra of finite type B_0, an ideal $J_{0}$ of B_0 and a B_0-module of finite type M_0 which is

a flat A_0-module and an A_0-module N_0 such that $B = B_{0} \otimes_{A_{0}} A$ , $J = J_{0} B$ , $M = M_{0} \otimes_{A_{0}} A$ and $N = N_{0} \otimes_{A_{0}} A$ (8.9.1, 8.5.11 and 11.2.7). Moreover, let $(A_{λ})$ be the family of sub-A_0-algebras of finite type of $A$ , so that $A = lim A_{λ}$ ; set $B_{λ} = B_{0} \otimes_{A_{0}} A_{λ}$ , $J_{λ} = J_{0} B_{λ}$ , $M_{λ} = M_{0} \otimes_{A_{0}} A_{λ}$ , $N_{λ} = N_{0} \otimes_{A_{0}} A_{λ}$ , $S_{λ} = Spec (A_{λ})$ , $X_{λ} = Spec (B_{λ})$ , $Z_{λ} = V (J_{λ})$ ; if $p_{λ} : X \to X_{λ}$ is the canonical projection, one has therefore $Z = p_{λ}^{- 1} (Z_{λ})$ , and $X - Z = p_{λ}^{- 1} (X_{λ} - Z_{λ})$ . As $X_{λ} - Z_{λ}$ is quasi-compact and quasi-separated, one concludes (8.5.2) that if one sets $H_{λ} = M_{λ} \otimes N_{λ}$ and $H_{λ} = (M_{λ} \otimes N_{λ})^{\sim}$ , the homomorphisms $lim Γ (X_{λ}, H_{λ}) \to Γ (X, H)$ and $lim Γ (X_{λ} - Z_{λ}, H_{λ}) \to Γ (X - Z, H)$ are bijective; by virtue of the exactness of the functor lim, it will therefore suffice to prove that for every sufficiently large $λ$ , the canonical homomorphism $Γ (X_{λ}, H_{λ}) \to Γ (X_{λ} - Z_{λ}, H_{λ})$ is injective (resp. bijective). But for every $x \in Z$ , if $x_{λ}$ is the projection of $x$ in $Z_{λ}$ , one has $p ro f ((F_{f (x)})_{x}) = p ro f (((F_{λ})_{f_{λ} (x_{λ})})_{x_{λ}})$ by virtue of (4.2.7) and (6.7.1); to be reduced to demonstrating (19.9.8) in the case where $A$ is Noetherian, one must prove that, for $λ$ sufficiently large, one has $p ro f (((F_{λ})_{f_{λ} (x_{λ})})_{x_{λ}}) \geq 1$ (resp. $\geq 2$ ) for every point $x_{λ} \in Z_{λ}$ . Now, let $Z_{λ}^{1}$ be the set of points $x_{λ} \in Z_{λ}$ such that, for every generization $ξ$ of $x_{λ}$ in $Z_{λ} \cap (X_{λ})_{f_{λ} (x_{λ})}$ , one has $p ro f (((F_{λ})_{f_{λ} (ξ)})_{ξ}) \geq 1$ ; if $p_{μ λ} : Z_{μ} \to Z_{λ}$ is the canonical projection for $λ \leq μ$ , it follows from the hypothesis and from (19.9.5) that one has $Z_{λ} = p_{μ λ}^{- 1} (Z_{μ}^{1})$ and $Z_{λ}^{1} = p_{μ λ}^{- 1} (Z_{λ}^{1})$ ; on the other hand, it follows from (19.9.6) that $Z_{λ}^{1}$ is open in $Z_{λ}$ , hence our assertion follows from (8.3.4).

To finish the proof when $A$ is Noetherian, note that by virtue of the flatness hypothesis and of (6.3.1), one then has $p ro f_{Z} (H) \geq 1$ (resp. $p ro f_{Z} (H) \geq 2$ ); but the conclusion of (19.9.8) follows in this case from (5.10.2) (resp. (5.10.4)).

Remark (19.9.9).

The same proof, joined to the results of Chap. III, 3rd part on depth and local cohomology, allows one to establish the following generalization of (19.9.8):

Under the general conditions of (19.9.8), suppose that, for every $x \in Z$ , one has $p ro f ((F_{f (x)})_{x}) \geq k$ ; then the canonical homomorphism

                       H^i(X, ℋ) → H^i(X − Z, ℋ | (X − Z))

is bijective for $i \leq k - 2$ and injective for $i = k - 1$ (which, expressed by means of the general cohomological notion of depth introduced in Chap. III, is also written $p ro f_{Z} (H) \geq k$ ).

The proof, without separation hypothesis on the $g r_{k} (M)$ , is due to P. Deligne, whom we thank.

Éléments de Géométrie Algébrique — English