Exposé X. Application to the fundamental group

Throughout this Exposé, $X$ will denote a locally noetherian prescheme, $Y$ a closed part of $X$, $U$ a variable open neighborhood of $Y$ in $X$, and $\hat{X}$ the formal completion of $X$ along $Y$ (EGA I 10.8). For every prescheme $Z$, we denote by $\hat{E}t(Z)$ the category of étale coverings of $Z$, and by $L(Z)$ the category of locally free coherent Modules on $Z$.

1. Comparison of $\hat{E}t(\hat{X})$ and $\hat{E}t(Y)$

Let $I$ be an ideal of definition of $Y$ in $X$. Set, for every $n\in \mathbb{N}$, $Y_n=(Y,({\mathcal{O}}_X/I^{n+1})|Y)$. The $Y_n$ form a direct system of ordinary preschemes, or also of formal preschemes, by equipping the structure sheaves with the discrete topology. One knows (EGA I 10.6.2) that $\hat{X}$ is the direct limit, in the category of formal preschemes, of the direct system of the $Y_n$. One also knows (EGA I 10.13) that to give a formal $\hat{X}$-prescheme of finite type $R$ is the same as to give a direct system of $Y_n$-preschemes $R_n$ of finite type, such that $R_n\simeq (R_{n+1}){\times }_{(Y_{n+1})}(Y_n)$. Moreover, in order that $R$ be an étale covering of $\hat{X}$, it is necessary and sufficient that for every $n$, $R_n$ be an étale covering of $Y_n$. This said, it is easy to see that nilpotent elements do not matter for étale coverings (SGA 1 8.3), that is, that the base-change functor

$$\hat{E}t(Y_{n+1})⟶\hat{E}t(Y_n)$$

is an equivalence of categories for every $n\in \mathbb{N}$. Hence:

Proposition.

With the notation introduced above, the natural functor $\hat{E}t(\hat{X})\to \hat{E}t(Y)$ is an equivalence of categories (cf. SGA 1 8.4).

2. Comparison of $\hat{E}t(Y)$ and $\hat{E}t(U)$, for $U$ variable

We shall introduce two conditions from which the announced comparison theorem will follow easily. Let $X$ be a locally noetherian prescheme and let $Y$ be a closed part of $X$. One says that the pair $(X,Y)$ satisfies the Lefschetz condition, written $Lef(X,Y)$, if, for every open $U$ of $X$ containing $Y$ and every locally free coherent sheaf $E$ on $U$, the natural homomorphism

Γ(U, E) ⟶ Γ(X̂, Ê)

is an isomorphism.

One says that the pair $(X,Y)$ satisfies the effective Lefschetz condition, written $Leff(X,Y)$, if $Lef(X,Y)$ holds and if, moreover, for every locally free coherent sheaf $E$ on $\hat{X}$, there exist an open neighborhood $U$ of $Y$, a locally free coherent sheaf $E$ on $U$, and an isomorphism $\hat{E}\simeq E$.

These conditions are satisfied in two important examples:

Example.¹

Let $A$ be a noetherian ring and let $t\in \mathfrak{r}(A)$ be an $A$-regular element belonging to the radical $\mathfrak{r}(A)$ of $A$. Suppose that $A$ is a quotient of a regular local ring and that $A$ is complete for the $t$-adic topology (for example $A$ complete for the $\mathfrak{r}(A)$-adic topology). Set $X^{\prime }=\mathrm{Spec}(A)$ and $Y^{\prime }=V(t)$; further, set $x=\mathfrak{r}(A)$ and $X=X^{\prime }-x$, $Y=Y^{\prime }-x$. So $X$ is open in $X^{\prime }$ and $Y=X\cap Y^{\prime }$. Then:

1. If, for every prime ideal $\mathfrak{p}$ of $A$ such that $\dim A/\mathfrak{p}=1$ (i.e. for every closed point of $X$), one has $prof{A}_{\mathfrak{p}}⩾2$, then $Lef(X,Y)$ holds;
2. if, moreover, for every prime ideal $\mathfrak{p}$ of $A$ such that $t\in \mathfrak{p}$ and $\dim A/\mathfrak{p}=1$ (i.e. for every closed point of $Y$), one has $prof{A}_{\mathfrak{p}}⩾3$, then $Leff(X,Y)$ holds.

Let us first show that, for every open neighborhood $U$ of $Y$ in $X$, the complement of $U$ in $X$ is a union of a finite number of closed points (in $X$). Note that $U$ is open in $X$, hence in $X^{\prime }$, so $Z^{\prime }=X^{\prime }-U$ is closed.

Let $I$ be an ideal of definition of $Z^{\prime }$; it suffices to prove that $A/I$ is of dimension 1. But $Z^{\prime }\cap Y^{\prime }=x$, so $A/(I+(t))$ is artinian, whence the conclusion by the "Hauptidealsatz".

The first hypothesis is equivalent to: "for every prime ideal $\mathfrak{p}$ of $A$, $\mathfrak{p}\ne \mathfrak{r}(A)$, one has $prof{A}_{\mathfrak{p}}⩾3-\dim A/\mathfrak{p}$". Indeed, $A$ is a quotient of a regular ring, so one may apply VIII 2.3 to the prescheme $X^{\prime }$, to the closed part $x$, and to the coherent sheaf ${\mathcal{O}}_{X^{\prime }}$, observing that $c(\mathfrak{p})=\dim (A/\mathfrak{p})$ for $\mathfrak{p}\in U=X^{\prime }-x$ (since $x$ is the closed point of $X^{\prime }$).

Let $U$ be an open neighborhood of $Y$ in $X$ and let $E$ be a locally free ${\mathcal{O}}_U$-module. Set $Z=X-U$ and let $u:U\to X$ be the canonical immersion. We shall first prove that $u_{\ast }(E)$ is a coherent ${\mathcal{O}}_X$-Module, or what amounts to the same, that ${\mathcal{H}}_Z^i(E^{\prime })$ is coherent for $i=0,1$, where $E^{\prime }$ is a coherent extension of $E$ to $X$. To do this, one applies Theorem VIII 2.1 to the prescheme $X$, to the closed part $Z$, and to the coherent sheaf $E^{\prime }$. It suffices to verify that for every point $\mathfrak{p}\in U$ such that $c(\mathfrak{p})=1$, one has $prof{E}_{\mathfrak{p}}^{\prime }⩾1$, where we have set

c(𝔭) = codim({𝔭}̄ ∩ Z, {𝔭}̄).

Now if $\mathfrak{p}\in U$ and $c(\mathfrak{p})=1$, denoting again by $\mathfrak{p}$ the ideal of $A$ corresponding to $\mathfrak{p}$, one sees that $\dim A/\mathfrak{p}=2$, since the complement of $U$ is a union of a finite number of closed points and $A$ is a quotient of a regular ring. Moreover, $E$ is locally free, so for every $\mathfrak{p}\in SuppE$ one has $prof{E}_{\mathfrak{p}}=prof{\mathcal{O}}_{U,\mathfrak{p}}$. Finally, if $\mathfrak{p}\in U$ and $c(\mathfrak{p})=1$, one has

prof E′_𝔭 = prof E_𝔭 = prof 𝒪_{U,𝔭} = prof A_𝔭 ⩾ 3 − 2 = 1.

We must now prove that the natural homomorphism

Γ(U, E) ⟶ Γ(X̂, Ê)

is an isomorphism. Setting then $E=u_{\ast }(E)$, one notes that Ẽ is coherent and of depth $⩾2$ at every closed point of $X$. It follows that $R^i{f}_{\ast }(E)$ is coherent for $i=0,1$, where $f:X\to X^{\prime }$ denotes the canonical immersion of $X$ into $X^{\prime }=\mathrm{Spec}(A)$ (Exp. VIII). One then applies (IX 1.5) and concludes that

Γ(U, E) ⟶ Γ(X̂, Ê)

is an isomorphism, since $A$ is complete for the $t$-adic topology. One has a commutative diagram

       Γ(X, Ẽ) ────≃────→ Γ(U, E)
            ╲              ╱
             ╲            ╱
              ≃          ↓
               ╲        ╱
                ↘     ↙
                Γ(X̂, Ê)

whence the conclusion.

Now let $E$ be a locally free coherent sheaf on $\hat{X}$. If one has proved that $E$ is algebraizable, i.e. is isomorphic to the formal completion of a coherent ${\mathcal{O}}_X$-Module Ẽ, it is easy to see that Ẽ is locally free in a neighborhood of $Y$, hence to prove $Leff(X,Y)$. Let ${\hat{X}}^{\prime }$ be the formal spectrum of $A$ for the $t$-adic topology, which is identified with the formal completion of $X^{\prime }$ along $Y^{\prime }$. Denote by $f$ the canonical immersion of $X$ in $X^{\prime }$, by $f^{\prime }$ the canonical immersion of $Y$ in $Y^{\prime }$, and by $\hat{f}$ the morphism deduced by passing to the completions. In order that $E$ be algebraizable, it suffices that ${\hat{f}}_{\ast }(E)$ be a coherent ${\mathcal{O}}_{{\hat{X}}^{\prime }}$-Module, since $A$ is complete for the $t$-adic topology. Let $I=t{\mathcal{O}}_{\hat{X}}$; this is an ideal of definition of $\hat{X}$.

For every $n⩾0$, set $E_n=E/I^{n+1}E$. At every closed point $y\in Y$, the depth of $E_0$ is $⩾2$; indeed, $t$ is an $A$-regular element, so $prof{\mathcal{O}}_{Y_0,y}=prof{\mathcal{O}}_{X,y}-1⩾2$. One concludes that ${\hat{f}}_{\ast }(E)$ is coherent (IX 2.3). QED.

Example (Will allow comparison of the fundamental group of a projective variety and a hyperplane section).

Let $K$ be a field and let $X$ be a proper $K$-prescheme. Let $L$ be an ample invertible ${\mathcal{O}}_X$-Module. Let $t\in \Gamma (X,L)$ be an ${\mathcal{O}}_X$-regular element, which means that, for every open $U$ and every isomorphism $u:L|U\to {\mathcal{O}}_U$, $u(t)$ is a non-zero-divisor in ${\mathcal{O}}_U$ (a condition that does not depend on $u$). Let $Y=V(t)$ be the subscheme of $X$ of equation $t=0$.² Then:

1. If, for every closed point $x$ in $X$, one has $prof{\mathcal{O}}_{X,x}⩾2$, then $Lef(X,Y)$ holds;
2. if, moreover, for every closed point $y\in Y$ one has $prof{\mathcal{O}}_{X,y}⩾3$, then $Leff(X,Y)$ holds.

This example will be treated in detail in Exp. XII.

Let $S$ be a prescheme; one knows (EGA II 6.1.2) that the functor which to every finite flat covering $r:R\to S$ associates the ${\mathcal{O}}_X$-Algebra $r_{\ast }({\mathcal{O}}_R)$ induces an equivalence between the category of finite flat coverings of $S$ and the category of locally free coherent ${\mathcal{O}}_X$-Algebras. Let $U$ be an open neighborhood of $Y$, and let $r:R\to U$ be a finite flat covering of $U$. Let $\hat{R}$ be the finite flat covering of $\hat{X}$ deduced from it by base change. One has ${\hat{r}}_{\ast }({\mathcal{O}}_{\hat{R}})\simeq r_{\ast }\widehat{{\mathcal{O}}_R}$.

Suppose then that $Lef(X,Y)$ holds. This implies that, for every $U$, the inverse image functor

$$L(U)⟶L(\hat{X})$$

is fully faithful. Indeed, let $E$ and $F$ be two locally free coherent ${\mathcal{O}}_U$-Modules; $\mathrm{Hom}(E,F)$ is also coherent and locally free. By hypothesis the natural map

Γ_U(Hom(E, F)) ⟶ Γ_{X̂}(Hom(E, F)̂)

is an isomorphism, whence the conclusion, since Hom commutes with ̂ since everything is locally free. Now the ̂ commutes with the tensor product, from which one deduces that the functor which to every locally free coherent ${\mathcal{O}}_U$-Algebra $A$ associates the ${\mathcal{O}}_{\hat{X}}$-Algebra Â is fully faithful. Better, if $E$ is a locally free coherent ${\mathcal{O}}_U$-Module, there is a bijective correspondence between the commutative ${\mathcal{O}}_{\hat{X}}$-Algebra structures on Ê.

Proposition.

Let $X$ be a locally noetherian prescheme and let $Y$ be a closed part of $X$. Let $\hat{X}$ be the formal completion of $X$ along $Y$. For every open $U$ of $X$, $U\supset Y$, denote by L_U (resp. P_U, resp. E_U) the functor which to every locally free coherent ${\mathcal{O}}_U$-Module (resp. every finite flat covering of $U$, resp. every étale covering of $U$) associates its inverse image by $\hat{X}\to X$.

1. If $Lef(X,Y)$ holds, then for every open neighborhood $U$ of $Y$, the functors L_U, P_U, and E_U are fully faithful.

1. If $Leff(X,Y)$ holds, then for every locally free coherent ${\mathcal{O}}_{\hat{X}}$-Module $E$ (resp. ...), there exist an open $U$ and a locally free coherent ${\mathcal{O}}_U$-Module Ẽ (resp. ...), such that $L_U(\tilde{E})\simeq E$ (resp. ...).

(i) Has been seen.

(ii) Follows from (i) and from the hypothesis, at least for L_U and P_U. Moreover, if $R$ is an étale covering of $\hat{X}$, there exist an open neighborhood $U$ of $Y$ in $X$ and a finite flat covering $R^{\prime }$ of $U$ such that ${\hat{R}}^{\prime }\simeq R$. From it one deduces a covering $R^{\prime \prime }$ of $Y$ which is étale by 1.1, so $R^{\prime }$ is étale in a neighborhood $U^{\prime }$ of $Y$. QED.

Corollary.

If $Lef(X,Y)$ holds, then in order that a finite flat covering $R$ of an open neighborhood $U$ of $Y$ be connected, it is necessary and sufficient that $R{\times }_U\hat{X}$ be connected. In particular, in order that $Y$ be connected, it is necessary and sufficient that the open neighborhood $U$ of $Y$ be connected, or again that $\hat{X}$ be connected.

Indeed, in order that a ringed space in local rings $(X,{\mathcal{O}}_X)$ be connected, it is necessary and sufficient that $\Gamma (X,{\mathcal{O}}_X)$ not be a direct product of two non-zero rings. Now one has

Γ(U, r_*(𝒪_R)) ≃ Γ(X̂, r̂_*(𝒪_{R̂}))

by $Lef(X,Y)$.

Corollary.

If $Lef(X,Y)$ holds, then for every $U$, the functor

$$\hat{E}t(U)⟶\hat{E}t(Y)$$

is fully faithful. If $Leff(X,Y)$ holds, then for every étale covering $R$ of $Y$, there exist an open neighborhood $U$ of $Y$ and a covering $R^{\prime }$ of $U$ such that $R^{\prime }{\times }_{U}Y\simeq R$.

Corollary.³

If $Lef(X,Y)$ holds and $Y$ is connected, then every open neighborhood $U$ of $Y$ is connected and the natural homomorphism ${\pi }_1(Y)\to {\pi }_1(U)$ is surjective. If, moreover, $Leff(X,Y)$ holds, the natural homomorphism

π₁(Y) ⟶ lim_{←, U} π₁(U)

is an isomorphism. (N.B. One assumes that a "base-point" has been chosen in $Y$, which one also takes as base-point in $X$, for the definition of the fundamental groups.)

All of this follows trivially from Proposition 1.1 and Proposition 2.3.

3. Comparison of ${\pi }_1(X)$ and ${\pi }_1(U)$

Definition.

Let $X$ be a prescheme and $Z$ a closed part of $X$. Set $U=X-Z$. One says that the pair $(X,Z)$ is pure if, for every open $V$ of $X$, the functor

Êt(V) ⟶ Êt(V ∩ U)
V′ ↦ V′ ×_V (V ∩ U)

is an equivalence of categories.⁴

Definition.

Let $A$ be a noetherian local ring. Set $X=\mathrm{Spec}A$. Let $\mathfrak{r}(A)$ be the radical of $A$ and let $x=\mathfrak{r}(A)$ be the closed point of $X$. One says that $A$ is pure if the pair (X, {x}) is.

We leave to the reader the task of not proving the following proposition:

Proposition.

Let $X$ be a locally noetherian prescheme and let $Z$ be a closed part of $X$. In order that the pair $(X,Z)$ be pure it is necessary and sufficient that, for every $z\in Z$, the ring ${\mathcal{O}}_{X,z}$ be pure.⁵

This said, the following theorem is the essential result of this number:

Theorem (Purity theorem).⁶

1. A regular noetherian local ring of dimension $⩾2$ is pure (Zariski–Nagata purity theorem).
2. A noetherian local ring of dimension $⩾3$ which is a complete intersection is pure.

Recall that one says that a local ring is a complete intersection if there exist a regular noetherian local ring $B$ and a $B$-regular sequence $(t_1,\cdots ,t_k)$ of elements of the radical $\mathfrak{r}(B)$ of $B$ such that

$$A\simeq B/(t_1,\cdots ,t_k).$$

In this connection, let us remark that it would be less ambiguous to say that $A$ is an absolute complete intersection, by opposition with the situation, already encountered, in which $X$ is a locally noetherian prescheme (which need not be regular) and $Y$ is a closed part of $X$, of which one says that it is "locally set-theoretically a complete intersection in $X$".

Let us first prove a few lemmas.

Lemma.

Let $X$ be a locally noetherian prescheme and let $U$ be an open part of $X$. Set $Z=X-U$. Let $i:U\to X$ be the canonical immersion of $U$ into $X$. The following conditions are equivalent:

1. For every open $V$ of $X$, if one sets $V^{\prime }=V\cap U$, the functor $F\mapsto F|V^{\prime }$ from the category of locally free coherent ${\mathcal{O}}_V$-Modules to the category of locally free coherent ${\mathcal{O}}_{V^{\prime }}$-Modules is fully faithful;
2. the natural homomorphism ${\mathcal{O}}_X\to i_{\ast }({\mathcal{O}}_U)$ is an isomorphism;
3. for every $z\in Z$, one has $prof{\mathcal{O}}_{X,z}⩾2$.

One has already seen (III 3.3) the equivalence of (ii) and (iii). Let us show that (ii) implies (i). Let $F$ and $G$ be two locally free coherent ${\mathcal{O}}_V$-Modules; $\mathrm{Hom}(F,G)$ is also one, so $\mathrm{Hom}(F,G)\to i_{\ast }(\mathrm{Hom}(F|V^{\prime },G|V^{\prime }))$ is an isomorphism, so $\mathrm{Hom}(F,G)\simeq \mathrm{Hom}(F|V^{\prime },G|V^{\prime })$. Conversely, one takes $F=G={\mathcal{O}}_X$ and applies (i) to every open $V$ of $X$.

Here is a useful "descent lemma":

Lemma.

Let $X$ be a locally noetherian prescheme and let $Z$ be a closed part of $X$. Set $U=X-Z$. Suppose that the homomorphism ${\mathcal{O}}_X\to i_{\ast }({\mathcal{O}}_U)$ is an isomorphism. Let $f:X_1\to X$ be a faithfully flat and quasi-compact morphism. Set $Z_1=f^{-1}(Z)$. If the pair $(X_1,Z_1)$ is pure, then so is $(X,Z)$.

Note that the hypothesis ${\mathcal{O}}_X\simeq i_{\ast }({\mathcal{O}}_U)$ is preserved by flat extension of the base, since $i$ is a quasi-compact morphism and, in that case, direct image commutes with inverse image. Now this hypothesis implies that the functor

$$\hat{E}t(V)⟶\hat{E}t(U\cap V)$$

defined by

V′ ↦ V′ ×_V (V ∩ U)

is fully faithful, as shown by the interpretation of an étale covering in terms of locally free coherent Algebras. It remains to prove effectivity.

One can, for example, introduce the square $X_2$ and the cube $X_3$ of $X_1$ over $X$ and observe that a faithfully flat and quasi-compact morphism is a morphism of universal effective descent for the fibered category of étale coverings, above the category of preschemes. The conclusion is formal from there.⁷

Remark.

We have shown in passing that if ${\mathcal{O}}_X\to i_{\ast }({\mathcal{O}}_U)$ is an isomorphism, then $X$ is connected if and only if $U$ is, and then ${\pi }_1(U)\to {\pi }_1(X)$ is surjective.

Corollary.

Let $A$ be a noetherian local ring. Suppose that $profA⩾2$. Then if Â is pure, $A$ is pure.

Follows from Lemma 3.5 and Lemma 3.6.

The following lemma is the essential point in the proof of the purity theorem:

Lemma.

Let $A$ be a noetherian local ring and let $t\in \mathfrak{r}(A)$ be an $A$-regular element. Suppose that $A$ is complete for the $t$-adic topology and is, moreover, a quotient of a regular local ring (for example $A$ complete). Set $B=A/tA$.

1. If for every prime ideal $\mathfrak{p}$ of $A$ such that $\dim A/\mathfrak{p}=1$, one has $prof{A}_{\mathfrak{p}}⩾2$, then $B$ pure implies $A$ pure.
2. If for every prime ideal $\mathfrak{p}$ of $A$ such that $\dim A/\mathfrak{p}=1$, one has $prof{A}_{\mathfrak{p}}⩾2$, if $A_{\mathfrak{p}}$ is pure when $t\notin \mathfrak{p}$, and if⁸ $prof{A}_{\mathfrak{p}}⩾3$ when $t\in \mathfrak{p}$, then $A$ pure implies $B$ pure.

Let $X^{\prime }=\mathrm{Spec}(A)$ and $Y^{\prime }=V(t)$, which one identifies with the spectrum of $B$. Let $x=\mathfrak{r}(A)$, and set $X=X^{\prime }-x$ and $Y=Y^{\prime }-x=X\cap Y^{\prime }$. Denote by ${\hat{X}}^{\prime }$ the formal spectrum of $A$ for the $t$-adic topology, which is identified with the formal completion of $X^{\prime }$ along $Y^{\prime }$.

Since $A$ is complete for the $t$-adic topology, one notes that $\hat{E}t(X^{\prime })\to \hat{E}t({\hat{X}}^{\prime })$ is an equivalence of categories. Likewise $\hat{E}t({\hat{X}}^{\prime })\to \hat{E}t(Y^{\prime })$ by Proposition 1.1, so $\hat{E}t(X^{\prime })\to \hat{E}t(Y^{\prime })$ is an equivalence of categories.

Let us show (i). Consider the diagrams

   X′  ←──  X                    Êt(X′)  ──a──→  Êt(X)
   ↑        ↑                       │              │
   │        │                       c              b
   │        │                       ↓              ↓
   Y′  ←──  Y                    Êt(Y′)  ──d──→  Êt(Y)

We have just seen that $c$ is an equivalence; $d$ is also one by the hypothesis that $B$ is pure; and finally $b$ is fully faithful as seen in Example 2.1, cf. 2.3 (i).

Let us show (ii). This time one assumes that $A$ is pure, so $a$ is an equivalence; likewise $c$. Let us see that $b$ is an equivalence. By Example 2.1 we know that $Leff(X,Y)$ holds, so $b$ is already fully faithful; let us prove that it is essentially surjective. One uses 2.3 (ii), noting that if $U$ is an open neighborhood of $Y$ in $X$, the complement of $U$ in $X$ is a union of a finite number of closed points; the pair $(X,X-U)$ is thus pure by Proposition 3.3, since at such a point $\mathfrak{p}$, ${\mathcal{O}}_{X,\mathfrak{p}}=A_{\mathfrak{p}}$ is pure by hypothesis. Whence the conclusion.

Proof of the purity theorem.

Let us first prove (i) by induction on the dimension. Let $A$ be a noetherian local ring of dimension 2. Set $X^{\prime }=\mathrm{Spec}(A)$, $x=\mathfrak{r}(A)$, $X=X^{\prime }-x$. One has $profA=2$. One may therefore apply Lemma 3.5 to the pair $(X^{\prime },x)$, and so $\hat{E}t(X^{\prime })\to \hat{E}t(X)$ is fully faithful. Let now $r:R\to X$ be an étale covering defined by a locally free coherent and étale ${\mathcal{O}}_X$-Algebra $A=r_{\ast }({\mathcal{O}}_R)$. Denote by $i:X\to X^{\prime }$ the canonical immersion of $X$ into $X^{\prime }$. I claim that $i_{\ast }(A)=B$ is a coherent ${\mathcal{O}}_{X^{\prime }}$-Algebra. Indeed, it suffices to apply the "finiteness theorem" VIII 2.3. I claim that this algebra is of depth $⩾2$ at $x$. Indeed, it is the direct image of an ${\mathcal{O}}_X$-Module, with $X=X^{\prime }-x$. Since $A$ is a regular ring of dimension 2, one has $dpB+profB=\dim A=2$, where dp B denotes the projective dimension of $B$. So $dpB=0$, hence $B$ is projective, hence free. It follows that $B$ defines a finite flat covering of $X^{\prime }=\mathrm{Spec}(A)$. The set of points of $X^{\prime }$ where this covering is not étale is a closed part of $X^{\prime }$ whose equation is a principal ideal: the discriminant ideal of $B/A$. Now, by construction, this closed set is contained in $x=\mathfrak{r}(A)$, hence is empty since $\dim A=2$.

Let $A$ be a regular noetherian local ring, $\dim A=n⩾3$. Suppose (i) proved for rings of dimension $<n$. To prove that $A$ is pure, one may assume $A$ complete by 3.8. Let $t\in \mathfrak{r}(A)$ whose image in $\mathfrak{r}(A)/\mathfrak{r}(A{)}^2$ is nonzero. Then $B=A/tA$ is a regular noetherian local ring of dimension $n-1$, hence is pure, since $n-1⩾2$. One concludes by Lemma 3.9 (i), which is applicable since $A$ is complete.

Let us show (ii). Let $A$ be a noetherian local ring of dimension $⩾3$. Suppose that there exist a regular noetherian local ring $B$ and a $B$-sequence $(t_1,\cdots ,t_k)$ such that $A\simeq B/(t_1,\cdots ,t_k)$. Let us prove that $A$ is pure, by induction on $k$. If $k=0$, one knows it by (i). Suppose $k⩾1$ and the result acquired for $k^{\prime }<k$. By Corollary 3.8⁹ one may assume that $A$ (hence also $B$) is complete. Set $C=B/(t_1,\cdots ,t_{k-1})$, so $A\simeq C/t_{k}C$ and $t_k$ is $C$-regular. By the induction hypothesis one knows that $C$ is pure; it suffices to prove that Lemma 3.9 (ii) is applicable. Notation: the $A$ and $B$ of the lemma become $C$ and $A$. One has $\dim C⩾4$, so for every prime ideal $\mathfrak{p}$ of $C$ such that $\dim C/\mathfrak{p}=1$, one has $prof{C}_{\mathfrak{p}}⩾3$. Moreover, $C_{\mathfrak{p}}$ is a complete intersection with $k^{\prime }⩽k-1$, hence is pure by the induction hypothesis. QED.

Theorem.

Let $X$ be a locally noetherian prescheme and let $Y$ be a closed part of $X$. Suppose that one has $Leff(X,Y)$ (cf. Examples 2.1 and 2.2). Suppose moreover that, for every open neighborhood $U$ of $Y$ and every $x\in X-U$, the local ring ${\mathcal{O}}_{X,x}$ is regular of dimension $⩾2$ or a complete intersection of dimension $⩾3$. Then

$${\pi }_0(Y)⟶{\pi }_0(X)$$

is a bijection, and if $X$ is connected

$${\pi }_1(Y)⟶{\pi }_1(X)$$

is an isomorphism.

There is nothing more to prove. One remarks that, in the two examples cited 2.1 and 2.2, the complement of $U$ is a union of a finite number of closed points, from which it follows that the hypothesis on the dimension of ${\mathcal{O}}_{X,x}$ is not a farce.