Éléments de Géométrie Algébrique — English

§17. Smooth, unramified, étale morphisms

In the present section, we take up again the notions studied in (0, 19), expressed by means of the geometric language of schemes and from the global point of view, for preschemes locally of finite presentation over a given base prescheme. Most of the results (with the exception of nos. 17.7, 17.8, 17.9, 17.13, and 17.16) in fact reduce to variants of properties already encountered in (0, 19). For more special results on étale morphisms, the reader will consult §18.

17.1. Formally smooth, formally unramified, formally étale morphisms

Definition (17.1.1).

Let $f:X\to Y$ be a morphism of preschemes. One says that $f$ is formally smooth (resp. formally unramified, resp. formally étale) if, for every affine scheme $Y^{\prime }$, every closed subscheme Y_0 of $Y^{\prime }$ defined by a nilpotent Ideal $\mathcal{J}$ of ${\mathcal{O}}_{Y^{\prime }}$, and every morphism $Y^{\prime }\to Y$, the map

  (17.1.1.1)    Hom_Y(Y', X) → Hom_Y(Y_0, X)

deduced from the canonical injection $Y_0\to Y^{\prime }$, is surjective (resp. injective, resp. bijective).

One also says in that case that $X$ is formally smooth (resp. formally unramified, resp. formally étale) over $Y$.

It is clear that to say that $f$ is formally étale signifies that it is at once formally smooth and formally unramified.

Remarks (17.1.2). — (i) Suppose that $Y=\mathrm{Spec}(A)$ and $X=\mathrm{Spec}(B)$ are affine, so that $f$ comes from a ring homomorphism $\varphi :A\to B$. By virtue of (0, 19.3.1 and 0, 19.10.1), to say that $f$ is formally smooth (resp. formally unramified, resp. formally étale) signifies that $\varphi$ makes $B$ into a formally smooth (resp. formally unramified, resp. formally étale) $A$-algebra for the discrete topologies on $A$ and $B$.

(ii) To verify that $f$ is formally smooth (resp. formally unramified, resp. formally étale), one can, in the definition (17.1.1), restrict to the case where ${\mathcal{J}}^2=0$. Indeed, if $f$ verifies the corresponding condition of definition (17.1.1) in this particular case, and if one has ${\mathcal{J}}^n=0$, one considers the closed subscheme $Y_j^{\prime }$ of $Y^{\prime }$ defined by the Ideal ${\mathcal{J}}^{j+1}$ for $0⩽j⩽n-1$, so that $Y_j^{\prime }$ is a closed subscheme of $Y_{j+1}^{\prime }$ defined by an Ideal of square zero; the hypothesis implies that each of the maps

  Hom_Y(Y'_{j+1}, X) → Hom_Y(Y'_j, X)    (0 ⩽ j ⩽ n − 1)

is surjective (resp. injective, resp. bijective); by composition, one concludes that it is the same for (17.1.1.1).

(iii) One will note that the properties of the morphism $f$ defined in (17.1.1) are properties of the representable functor $(0_{III},\mathrm{8.1.8})$

$$Y^{\prime }\mapsto {\mathrm{Hom}}_Y(Y^{\prime },X)$$

from the category of $Y$-preschemes to the category of sets; they retain a meaning for any contravariant functor having the same source and target categories, representable or not.

(iv) Suppose that the morphism $f$ is formally unramified (resp. formally étale); consider an arbitrary $Y$-prescheme $Z$ and a closed sub-prescheme Z_0 of $Z$ defined by a locally nilpotent Ideal $\mathcal{J}$ of ${\mathcal{O}}_Z$. Then the map

  (17.1.2.1)    Hom_Y(Z, X) → Hom_Y(Z_0, X)

deduced from the canonical injection $Z_0\to Z$, is still injective (resp. bijective). Indeed, let $(U_{\alpha })$ be an affine open cover of $Z$ such that the Ideals $\mathcal{J}|U_{\alpha }$ are nilpotent, and for each $\alpha$, let $U_{\alpha ,0}$ be the inverse image of $U_{\alpha }$ in Z_0, which is the closed sub-prescheme of $U_{\alpha }$ defined by $\mathcal{J}|U_{\alpha }$. Let $f_0:Z_0\to X$ be a $Y$-morphism; by hypothesis, for each $\alpha$, there is at most one (resp. one and only one) $Y$-morphism $f_{\alpha }:U_{\alpha }\to X$ whose restriction to $U_{\alpha ,0}$ is equal to $f_0|U_{\alpha ,0}$. One concludes at once that if $f_{\alpha }$ and $f_{\beta }$ are defined, then, for every affine open $V\subset U_{\alpha }\cap U_{\beta }$, one has $f_{\alpha }|V=f_{\beta }|V$, since the restrictions of these morphisms to the inverse image V_0 of $V$ in Z_0 coincide. Hence there is at most one (resp. one and only one) $Y$-morphism $f:Z\to X$ whose restriction to Z_0 coincides with $f_0$.

Proposition (17.1.3).

(i) A monomorphism of preschemes is formally unramified; an open immersion is formally étale.

(ii) The composite of two formally smooth (resp. formally unramified, resp. formally étale) morphisms is formally smooth (resp. formally unramified, resp. formally étale).

(iii) If $f:X\to Y$ is a formally smooth (resp. formally unramified, resp. formally étale) $S$-morphism, then so is $f_{(S^{\prime })}:X_{(S^{\prime })}\to Y_{(S^{\prime })}$ for every extension $S^{\prime }\to S$ of the base prescheme.

(iv) If $f:X\to X^{\prime }$ and $g:Y\to Y^{\prime }$ are two formally smooth (resp. formally unramified, resp. formally étale) $S$-morphisms, then so is $f{\times }_{S}g:X{\times }_{S}Y\to X^{\prime }{\times }_S{Y}^{\prime }$.

(v) Let $f:X\to Y$, $g:Y\to Z$ be two morphisms; if $g\circ f$ is formally unramified, then so is $f$.

(vi) If $f:X\to Y$ is a formally unramified morphism, then so is f_red : X_red → Y_red.

By virtue of (I, 5.5.12), it suffices to prove (i), (ii), and (iii). The two assertions of (i) are trivial. To prove (ii), consider two morphisms $f:X\to Y$, $g:Y\to Z$, an affine scheme $Z^{\prime }$, a closed subscheme $Z_0^{\prime }$ of $Z^{\prime }$ defined by a nilpotent Ideal, and a morphism $Z^{\prime }\to Z$. Suppose $f$ and $g$ are formally smooth, and consider a $Z$-morphism

$u_0:Z_0^{\prime }\to X$; the hypothesis on $g$ implies that there exists a $Z$-morphism $v:Z^{\prime }\to Y$ such that $f\circ u_0=v\circ j$ (where $j:Z_0^{\prime }\to Z^{\prime }$ is the canonical injection); the hypothesis on $f$ then implies that there exists a morphism $u:Z^{\prime }\to X$ such that $f\circ u=v$ and $u\circ j=u_0$, hence $(g\circ f)\circ u$ is equal to the given morphism $Z^{\prime }\to Z$ and $u\circ j=u_0$, which proves that $g\circ f$ is formally smooth; one reasons similarly when one supposes $f$ and $g$ formally unramified.

Finally, to prove (iii), set $X^{\prime }=X_{(S^{\prime })}$, $Y^{\prime }=Y_{(S^{\prime })}$, $f^{\prime }=f_{(S^{\prime })}$; consider an affine scheme Y'', a closed subscheme $Y_0^{\prime \prime }$ of Y'' defined by a nilpotent Ideal, and a morphism $q:Y^{\prime \prime }\to Y^{\prime }$ making Y'' into a $Y^{\prime }$-prescheme; one knows then (I, 3.3.8) that ${\mathrm{Hom}}_{Y^{\prime }}(Y^{\prime \prime },X^{\prime })$ canonically identifies with ${\mathrm{Hom}}_Y(Y^{\prime \prime },X)$ and ${\mathrm{Hom}}_{Y^{\prime }}(Y_0^{\prime \prime },X^{\prime })$ with ${\mathrm{Hom}}_Y(Y_0^{\prime \prime },X)$, and the conclusion then results immediately from definition (17.1.1).

One will note that a closed immersion is not necessarily a formally smooth morphism.

Proposition (17.1.4).

Let $f:X\to Y$, $g:Y\to Z$ be two morphisms, and suppose $g$ formally unramified. Then, if $g\circ f$ is formally smooth (resp. formally étale), so is $f$.

Indeed, let $Y^{\prime }$ be an affine scheme, $Y_0^{\prime }$ a closed subscheme of $Y^{\prime }$ defined by a nilpotent Ideal, $h:Y^{\prime }\to Y$ a morphism, $j:Y_0^{\prime }\to Y^{\prime }$ the canonical injection, $u_0:Y_0^{\prime }\to X$ a $Y$-morphism, hence such that $f\circ u_0=h\circ j$. Suppose $g\circ f$ formally smooth; then there exists a morphism $u:Y^{\prime }\to X$ such that $u\circ j=u_0$ and $(g\circ f)\circ u=g\circ h$. But these relations imply that $f\circ u$ and $h$ are two $Z$-morphisms from $Y^{\prime }$ to $Y$ such that $(f\circ u)\circ j=h\circ j$; by virtue of the hypothesis that $g$ is formally unramified, one deduces that $f\circ u=h$, in other words $u$ is a $Y$-morphism; hence $f$ is formally smooth. Taking (17.1.3, (v)) into account, this proves the proposition.

Corollary (17.1.5).

Suppose $g$ formally étale; then, for $g\circ f$ to be formally smooth (resp. formally unramified, resp. formally étale), it is necessary and sufficient that $f$ be so.

This results from (17.1.4) and (17.1.3, (ii) and (v)).

Proposition (17.1.6).

Let $f:X\to Y$ be a morphism of preschemes.

(i) Let $(U_{\alpha })$ be an open cover of $X$ and, for each $\alpha$, let $i_{\alpha }:U_{\alpha }\to X$ be the canonical injection. For $f$ to be formally smooth (resp. formally unramified, resp. formally étale), it is necessary and sufficient that each of the morphisms $f\circ i_{\alpha }$ be so.

(ii) Let $(V_{\lambda })$ be an open cover of $Y$. For $f$ to be formally smooth (resp. formally unramified, resp. formally étale), it is necessary and sufficient that each of the restrictions $f^{-1}(V_{\lambda })\to V_{\lambda }$ of $f$ be so.

We first note that (ii) is a consequence of (i): indeed, if $j_{\lambda }:V_{\lambda }\to Y$ and $i_{\lambda }:f^{-1}(V_{\lambda })\to X$ are the canonical injections, the restriction $f_{\lambda }:f^{-1}(V_{\lambda })\to V_{\lambda }$ of $f$ is such that $j_{\lambda }\circ f_{\lambda }=f\circ i_{\lambda }$; if $f$ is formally smooth (resp. formally unramified), so is $f\circ i_{\lambda }$ since $i_{\lambda }$ is formally étale (17.1.3); but since $j_{\lambda }$ is formally étale, this implies that $f_{\lambda }$ is formally smooth (resp. formally unramified) by virtue of (17.1.5). Conversely, if all the $f_{\lambda }$ are formally smooth (resp. formally unramified), so are the $j_{\lambda }\circ f_{\lambda }$ (17.1.3), hence also $f$ by virtue of (i).

If one takes into account the fact that the $i_{\alpha }$ are formally étale, everything reduces to proving that if the $f\circ i_{\alpha }$ are formally smooth (resp. formally unramified), then so is $f$.

Let then $Y^{\prime }$ be an affine scheme, $Y_0^{\prime }$ a closed subscheme of $Y^{\prime }$ defined by a nilpotent Ideal $\mathcal{J}$, which one may suppose such that ${\mathcal{J}}^2=0$ (17.1.2, (ii)), and finally let $q:Y^{\prime }\to Y$ be a morphism. Suppose given a $Y$-morphism $u_0:Y_0^{\prime }\to X$; designate by $W_{\alpha }^{\prime }$ (resp. $W_{\alpha ,0}^{\prime }$) the prescheme induced by $Y^{\prime }$ (resp. $Y_0^{\prime }$) on the open $u_0^{-1}(U_{\alpha })$ (recall that $Y^{\prime }$ and $Y_0^{\prime }$ have the same underlying topological space). Suppose first that the $f\circ i_{\alpha }$ are formally unramified, and let us show that, if $u^{\prime }$ and u'' are two $Y$-morphisms from $Y^{\prime }$ to $X$ whose restrictions to $Y_0^{\prime }$ coincide, then $u^{\prime }=u^{\prime \prime }$. Indeed, taking (17.1.2, (iv)) into account, the hypothesis that the $f\circ i_{\alpha }$ are unramified implies that for each $\alpha$, one has $u^{\prime }|W_{\alpha }^{\prime }=u^{\prime \prime }|W_{\alpha }^{\prime }$, since the restrictions of these two $Y$-morphisms to $W_{\alpha ,0}^{\prime }$ coincide. Whence the conclusion in this case.

Suppose now all the $f\circ i_{\alpha }$ formally smooth and let us prove that there exists a $Y$-morphism $u:Y^{\prime }\to X$ of which $u_0$ is the restriction to $Y_0^{\prime }$. Now, since $Y^{\prime }$ is an affine scheme, one can apply (16.5.17), whose hypotheses are satisfied, and whose conclusion proves precisely the existence of $u$.

One can therefore say that the notions introduced in (17.1.1) are local on $X$ and on $Y$, which always allows one, by virtue of (17.1.2, (i)), to reduce to the study of formally smooth (resp. formally unramified, resp. formally étale) algebras.

17.2. General differential properties

Proposition (17.2.1).

For a morphism $f:X\to Y$ to be formally unramified, it is necessary and sufficient that ${\Omega }_{X/Y}^1=0$ (which is also written ${\Omega }_{X/Y}=0$ (16.3.1)).

Taking (17.1.6) into account, one is reduced to the case where $Y=\mathrm{Spec}(A)$ and $X=\mathrm{Spec}(B)$ are affine, and the conclusion then results from (0, 20.7.4) and the interpretation of ${\Omega }_{X/Y}$ in this case (16.3.7).

Corollary (17.2.2).

Let $f:X\to Y$, $g:Y\to Z$ be two morphisms. For $f$ to be formally unramified, it is necessary and sufficient that the canonical homomorphism (16.4.19)

$$f\ast ({\Omega }_{Y/Z}^1)\to {\Omega }_{X/Z}^1$$

be surjective.

This is an immediate consequence of (17.2.1) and of the exact sequence (16.4.19.1).

Proposition (17.2.3).

Let $f:X\to Y$ be a formally smooth morphism.

(i) The ${\mathcal{O}}_X$-Module ${\Omega }_{X/Y}^1$ is locally projective (16.10.1). If $f$ is locally of finite type, ${\Omega }_{X/Y}^1$ is locally free of finite type.

(ii) For every morphism $g:Y\to Z$, the sequence (16.4.19) of ${\mathcal{O}}_X$-Modules

$$(\mathrm{17.2.3.1})0\to f\ast ({\Omega }_{Y/Z}^1)\to {\Omega }_{X/Z}^1\to {\Omega }_{X/Y}^1\to 0$$

is exact; moreover, for every $x\in X$, there exists an open neighbourhood $U$ of $x$ such that the restrictions to $U$ of the homomorphisms of (17.2.3.1) form an exact and split sequence.

(i) One knows (16.3.9) that if $f$ is locally of finite type, ${\Omega }_{X/Y}^1$ is an ${\mathcal{O}}_X$-Module of finite type. To prove that, in every case, it is locally projective, one can restrict, by virtue of (17.1.6), to the case where $Y=\mathrm{Spec}(A)$ and $X=\mathrm{Spec}(B)$ are affine, and this results from the hypothesis on $f$ and from (0, 20.4.9 and 0, 19.2.1).

(ii) Here again, one can restrict to the case where $X$, $Y$, and $Z$ are affine (17.1.6), and the conclusion results in this case from the interpretation of the Modules figuring in the sequence (17.2.3.1) and from (0, 20.5.7).

Corollary (17.2.4).

If $f:X\to Y$ is a formally étale morphism, then, for every morphism $g:Y\to Z$, the canonical homomorphism of ${\mathcal{O}}_X$-Modules

$$f\ast ({\Omega }_{Y/Z}^1)\to {\Omega }_{X/Z}^1$$

is bijective.

This results from the exactness of the sequence (17.2.3.1) and from the fact that one then has ${\Omega }_{X/Y}^1=0$ (17.2.1).

Proposition (17.2.5).

Let $f:X\to Y$ be a morphism, $X^{\prime }$ a sub-prescheme of $X$ such that the composite morphism $X^{\prime }{\to }^{j}X{\to }^{f}Y$ (where $j$ is the canonical injection) is formally smooth. Then the sequence of ${\mathcal{O}}_{X^{\prime }}$-Modules (16.4.21)

  (17.2.5.1)    0 → 𝒩_{X'/X} → Ω^1_{X/Y} ⊗ 𝒪_{X'} → Ω^1_{X'/Y} → 0

is exact; moreover, for every $x\in X^{\prime }$, there exists an open neighbourhood $U$ of $x$ such that the restrictions to $U$ of the homomorphisms of (17.2.5.1) form an exact and split sequence.

Still by virtue of (17.1.6), one can restrict to the case where $Y=\mathrm{Spec}(A)$ and $X=\mathrm{Spec}(B)$ are affine, and $X^{\prime }=\mathrm{Spec}(B/\mathfrak{j})$, where $\mathfrak{j}$ is an ideal of $B$. Then the conormal sheaf ${\mathcal{N}}_{X^{\prime }/X}$ corresponds to the $B$-module $\mathfrak{j}/{\mathfrak{j}}^2$ (16.1.3), and the conclusion follows from (0, 20.5.14).

Proposition (17.2.6).

Let $X$, $Y$ be two preschemes, $f:X\to Y$ a morphism locally of finite type. The following conditions are equivalent:

a) $f$ is a monomorphism.

b) $f$ is radicial and formally unramified.

c) For every $y\in Y$, the fibre $f^{-1}(y)$ is empty or $k(y)$-isomorphic to $\mathrm{Spec}(k(y))$ (in other words, is reduced to a single point $x$ such that $k(y)\to {\mathcal{O}}_{X,x}/{\mathfrak{m}}_y{\mathcal{O}}_{X,x}$ is an isomorphism).

The fact that a) implies c) results from (8.11.5.1). It is clear that c) implies that $f$ is radicial; let us show that it also follows from c) that ${\Omega }_{X/Y}^1=0$, which will prove that c) implies b) (17.2.1). Note that the ${\mathcal{O}}_X$-Module ${\Omega }_{X/Y}^1$ is quasi-coherent of finite type (16.3.9). It therefore results from (I, 9.1.13.1) that, for $({\Omega }_{X/Y}^1{)}_x=0$, it is necessary and sufficient that if one puts $Y_1=\mathrm{Spec}(k(y))$, $X_1=f^{-1}(y)=X{\times }_Y{Y}_1$, one has $({\Omega }_{X_1/Y_1}^1{)}_x=0$; but since the morphism $f_1:X_1\to Y_1$ deduced from $f$ is formally unramified by virtue of hypothesis c) (17.1.3), the conclusion results from (17.2.1). Let us prove finally that b) implies a); for this, consider the diagonal morphism $g={\Delta }_f:X\to X{\times }_{Y}X$; since $f$ is radicial, $g$ is surjective (I, 8.7.1); on the other hand, ${\Omega }_{X/Y}^1$ is by definition the conormal sheaf $\mathcal{N}(g)$ of the immersion $g$ (16.3.1), and to say that $f$ is formally unramified therefore signifies that

$\mathcal{N}(g)=0$ (17.2.1). Moreover, $g$ is locally of finite presentation (1.4.3.1); hence the hypothesis $\mathcal{N}(g)=0$ implies that $g$ is an open immersion (16.1.10); being surjective, this immersion is an isomorphism, hence $f$ is a monomorphism (I, 5.3.8).

17.3. Smooth, unramified, étale morphisms

Definition (17.3.1).

One says that a morphism $f:X\to Y$ is smooth (resp. unramified, or net¹, resp. étale) if it is locally of finite presentation and formally smooth (resp. formally unramified, resp. formally étale).

One also says in that case that $X$ is smooth (resp. unramified or net, resp. étale) over $Y$.

We shall see further on (17.5.2) that this definition of a smooth morphism coincides with the one already given in (6.8.1); until then, it is the definition of (17.3.1) that we shall use exclusively.

It is clear that to say that $f$ is étale signifies that it is at once smooth and unramified.

Remarks (17.3.2). — (i) One will note that definition (17.3.1) can be expressed solely by means of the functor

$$Y^{\prime }\mapsto {\mathrm{Hom}}_Y(Y^{\prime },X)$$

considered in (17.1.2, (iii)), since to say that $f$ is locally of finite presentation amounts to saying that the preceding functor commutes with projective limits of affine schemes (8.14.2).

(ii) Let $A$ be a ring, $B$ an $A$-algebra. One says that $B$ is a smooth (resp. unramified, resp. étale) $A$-algebra if the corresponding morphism $\mathrm{Spec}(B)\to \mathrm{Spec}(A)$ is smooth (resp. unramified, resp. étale). It amounts to the same to say that $B$ is an $A$-algebra of finite presentation (1.4.6) and formally smooth (resp. formally unramified, resp. formally étale) for the discrete topologies.

(iii) It results from (17.1.6) and from the definition of a morphism locally of finite presentation (1.4.2) that the notions of smooth, unramified, and étale morphism are local on $X$ and on $Y$.

Proposition (17.3.3).

(i) An open immersion is étale. For an immersion to be unramified, it is necessary and sufficient that it be locally of finite presentation.

(ii) The composite of two smooth (resp. unramified, resp. étale) morphisms is smooth (resp. unramified, resp. étale).

(iii) If $f:X\to Y$ is a smooth (resp. unramified, resp. étale) $S$-morphism, then so is $f_{(S^{\prime })}:X_{(S^{\prime })}\to Y_{(S^{\prime })}$ for every extension $S^{\prime }\to S$ of the base prescheme.

(iv) If $f:X\to X^{\prime }$ and $g:Y\to Y^{\prime }$ are two smooth (resp. unramified, resp. étale) $S$-morphisms, then so is $f{\times }_{S}g:X{\times }_{S}Y\to X^{\prime }{\times }_S{Y}^{\prime }$.

(v) Let $f:X\to Y$, $g:Y\to Z$ be two morphisms; if $g$ is locally of finite type and $g\circ f$ is unramified, then $f$ is unramified.

This results immediately from (1.4.3) and (17.1.3).

Proposition (17.3.4).

Let $f:X\to Y$, $g:Y\to Z$ be two morphisms, and suppose $g$ unramified. Then, if $g\circ f$ is smooth (resp. unramified, resp. étale), so is $f$.

Indeed, since $g$ and $g\circ f$ are locally of finite presentation, so is $f$ (1.4.3, (v)); the conclusion thus results from (17.1.4) and (17.1.3, (v)).

Corollary (17.3.5).

Suppose $g$ étale; then, for $f$ to be smooth (resp. unramified, resp. étale), it is necessary and sufficient that $g\circ f$ be so.

This results from (17.3.4) and (17.3.3, (ii)).

Proposition (17.3.6).

Let $g:Y\to S$, $h:X\to S$ be two morphisms locally of finite presentation. For an $S$-morphism $f:X\to Y$ to be unramified, it is necessary and sufficient that the canonical homomorphism (16.4.19)

$$f\ast ({\Omega }_{Y/S}^1)\to {\Omega }_{X/S}^1$$

be surjective.

Since $f$ is then locally of finite presentation (1.4.3, (v)), the proposition results immediately from (17.2.2).

Definition (17.3.7).

Let $f:X\to Y$ be a morphism. One says that $f$ is smooth (resp. unramified, resp. étale) at a point $x\in X$ if there exists an open neighbourhood $U$ of $x$ in $X$ such that the restriction $f|U$ is a smooth (resp. unramified, resp. étale) morphism from $U$ to $Y$.

One also says in that case that $X$ is smooth (resp. unramified, resp. étale) over $Y$ at the point $x$.

Taking the remark (17.3.2, (iii)) into account, it amounts to the same to say that $f$ is a smooth (resp. unramified, resp. étale) morphism or that it is smooth (resp. unramified, resp. étale) at every point of $X$.

It is clear that the set of points of $X$ where a morphism $f:X\to Y$ is smooth (resp. unramified, resp. étale) is open in $X$.

Proposition (17.3.8).

For every prescheme $Y$ and every locally free ${\mathcal{O}}_Y$-Module $\mathcal{E}$ of finite type, the vector-bundle prescheme $\mathbf{V}(\mathcal{E})$ (II, 1.7.8) is a smooth $Y$-prescheme.

Indeed (17.3.2, (iii)), one can restrict to the case where $Y=\mathrm{Spec}(A)$ is affine and $\mathbf{V}(\mathcal{E})=\mathrm{Spec}(A[T_1,\cdots ,T_r])$; since $A[T_1,\cdots ,T_r]$ is an $A$-algebra formally smooth for the discrete topologies (0, 19.3.2) and of finite presentation, this proves the proposition (17.3.2, (ii)).

Corollary (17.3.9).

Under the hypotheses of (17.3.8), the projective-bundle prescheme $\mathbf{P}(\mathcal{E})$ (II, 4.1.1) is a smooth $Y$-prescheme.

One can again restrict to the case where $Y=\mathrm{Spec}(A)$ is affine and $\mathbf{P}(\mathcal{E})={\mathbf{P}}_A^r$. One knows then (II, 2.3.14) that one has a finite open cover of ${\mathbf{P}}_A^r$ by taking the $D_{+}(T_i)$ $(0⩽i⩽r)$, equal respectively to the spectra of the rings $S_{(T_i)}$, where one replaces $S$ by $A[T_0,T_1,\cdots ,T_r]$ and $f$ by $T_i$; but it follows at once from the definition of $S_{(f)}$ (II, 2.2.1) that this ring, in the case considered, is isomorphic to $A[T_0,\cdots ,T_{i-1},T_{i+1},\cdots ,T_r]$; hence the corollary results from (17.3.8).

17.4. Characterizations of unramified morphisms

Theorem (17.4.1).

Let $f:X\to Y$ be a morphism locally of finite presentation, $x$ a point of $X$. The following properties are equivalent:

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

b) The diagonal morphism ${\Delta }_f:X\to X{\times }_{Y}X$ is a local isomorphism at the point $x$.

b') If one sets ${\Delta }_f=(1_X,\theta )$, $Z=X{\times }_{Y}X$ and $z={\Delta }_f(x)$, the homomorphism ${\theta }_x:{\mathcal{O}}_{Z,z}\to {\mathcal{O}}_{X,x}$ is bijective.

b'') For every morphism $g:Y^{\prime }\to Y$, and every point $y^{\prime }\in Y^{\prime }$ over $y=f(x)$, every $Y^{\prime }$-section $s^{\prime }$ of $X^{\prime }=X{\times }_Y{Y}^{\prime }$ such that $x^{\prime }=s^{\prime }(y^{\prime })$ lies over $x$ is a local isomorphism at the point $y^{\prime }$.

c) One has $({\Omega }_{X/Y}^1{)}_x=0$.

d) The $k(y)$-prescheme $f^{-1}(y)$ is unramified over $k(y)$ at the point $x$.

d') The point $x$ is isolated in $X_y=f^{-1}(y)$ (in other words $(Er{r}_{III},20)$, the morphism $f$ is quasi-finite at the point $x$) and the ring ${\mathcal{O}}_{X_y,x}$ is a field, separable extension of $k(y)$.

d'') The ring ${\mathcal{O}}_{X_y,x}={\mathcal{O}}_{X,x}/{\mathfrak{m}}_y{\mathcal{O}}_{X,x}$ is a field, finite separable extension of $k(y)$.

e) The ring ${\mathcal{O}}_{X,x}$ is an ${\mathcal{O}}_{Y,y}$-algebra formally unramified for the discrete topologies.

Since $f$ is locally of finite type, the ${\mathcal{O}}_X$-Module ${\Omega }_{X/Y}^1$ is of finite type (16.3.9), so it amounts to the same to say that $({\Omega }_{X/Y}^1{)}_x=0$ or that there exists an open neighbourhood $U$ of $x$ such that ${\Omega }_{X/Y}^1|U=0$. Taking (17.2.1) into account, this proves the equivalence of a) and c). On the other hand, if one sets $A={\mathcal{O}}_{Y,y}$, $B={\mathcal{O}}_{X,x}$, one has $({\Omega }_{X/Y}^1{)}_x={\Omega }_{B/A}^1$ (16.4.5), and the equivalence of c) and e) therefore results from (0, 20.7.4).

Since d') involves only properties of the morphism $X_y\to \mathrm{Spec}(k(y))$, the equivalence of a) and d') will ipso facto imply that of d) and d'). On the other hand d') and d'') are equivalent, since it amounts to the same to say that ${\mathcal{O}}_{X_y,x}$ is a finite $k(y)$-algebra or that $x$ is an isolated point of $X_y$, since $X_y$ is a $k(y)$-prescheme locally of finite type (I, 6.4.4).

Let us now prove the equivalence of b) and b'). One can limit oneself to the case where $Y=\mathrm{Spec}(R)$ and $X=\mathrm{Spec}(S)$ are affine and $f$ of finite presentation; then one has $Z=\mathrm{Spec}(S{\otimes }_{R}S)$ and ${\Delta }_f$ corresponds to the canonical surjective homomorphism $S{\otimes }_{R}S\to S$, whose kernel $\mathfrak{j}$ is known to be an ideal of finite type (0, 20.4.4). If one sets $\mathcal{I}=\tilde{\mathfrak{j}}$, the ${\mathcal{O}}_Z$-Module ${\mathcal{O}}_X={\mathcal{O}}_Z/\mathcal{I}$ is thus of finite presentation, and the hypothesis that the homomorphism ${\theta }_x:{\mathcal{O}}_{Z,z}\to ({\mathcal{O}}_X{)}_z$ is bijective implies that, replacing $X$ if needed by an open neighbourhood of $x$, the homomorphism $\theta :{\mathcal{O}}_Z\to j_{\ast }({\mathcal{O}}_X)$ is itself bijective $(0_I,\mathrm{5.2.7})$. This therefore shows that b') implies b); the converse is evident.

On the other hand, the equivalence of b) and b'') results from (I, 5.3.7) without any finiteness hypothesis on $f$: the datum of a $Y^{\prime }$-section $s^{\prime }:Y^{\prime }\to X^{\prime }$ is equivalent to that of a

$Y$-morphism $h=g^{\prime }\circ s^{\prime }:Y^{\prime }\to X$ (where $g^{\prime }:X^{\prime }\to X$ is the canonical projection), so that $s^{\prime }=(1_{Y^{\prime }},h{)}_X$, and then the diagram

  (17.4.1.1)
                       Y' ──s'──→ X' ──→ Y' ×_Y X
                       │                    │
                       h                  𝟏_{Y'} × Δ_f
                       ↓                    ↓
                       X ──────Δ_f─────→ X ×_Y X

identifies $Y^{\prime }$ with the product of the $(X{\times }_{Y}X)$-preschemes $X$ and $X^{\prime }$. Consequently (I, 4.3.2), if ${\Delta }_f$ is a local isomorphism at the point $x$, $s^{\prime }$ is a local isomorphism at the point $y^{\prime }$ (since $x=h(y^{\prime })$), which proves that b) implies b''). The converse is obtained by applying b'') to the case where one takes $Y^{\prime }=X$, $y^{\prime }=x$, $g=f$, and $s^{\prime }={\Delta }_f$.

To complete the proof of (17.4.1), it suffices to prove the implications

  d'') ⇒ c) ⇒ b) ⇒ d'').

d'') ⇒ c): Since ${\Omega }_{X/Y}^1$ is an ${\mathcal{O}}_X$-Module of finite type, it results from Nakayama's lemma that the condition c) is equivalent to $({\Omega }_{X/Y}^1{)}_x{\otimes }_{{\mathcal{O}}_{X,x}}k(x)=0$, that is (16.4.5), $({\Omega }_{X_y/\mathrm{Spec}(k(y))}^1{)}_x=0$. One is therefore reduced to the case where $Y$ is the spectrum of a field $k$ and $X$ a $k$-algebraic prescheme. The hypothesis that ${\mathcal{O}}_{X,x}$ is a field $k^{\prime }$, finite extension of $k$, implies first that $x$ is closed in $X$ (I, 6.4.2), then that $x$ is a maximal point of the Noetherian prescheme $X$, hence is an isolated point of $X$. Replacing $X$ by the open set $x$ of $X$, one can therefore suppose that $X=\mathrm{Spec}(k^{\prime })$; but then the hypothesis that $k^{\prime }$ is a finite separable extension of $k$ implies ${\Omega }_{k^{\prime }/k}^1=0$ (0, 20.6.20), which proves c).

c) ⇒ b): One has seen above that one then has ${\Omega }_{X/Y}^1|U=0$ for an open neighbourhood $U$ of $x$ in $X$; the assertion b) results then from the definition of ${\Omega }_{X/Y}^1$ (16.3.1) and from (16.1.9).

b) ⇒ d''): Replacing $X$ by an open neighbourhood of $x$, one can suppose that ${\Delta }_f$ is an open immersion; if one designates by $f_y:X_y\to \mathrm{Spec}(k(y))$ the morphism deduced from $f$ by base change, ${\Delta }_{f_y}$ is then also an open immersion (I, 5.3.4), and since condition d'') concerns only the prescheme $X_y$, one sees that one can restrict to the case where $Y$ is the spectrum of a field $k$, $X$ the spectrum of a $k$-algebra $A$ of finite type; property d'') will be established if one proves that $A$ is a finite separable $k$-algebra, such an algebra being a direct composite of finite separable extensions of $k$. If $K$ is an algebraically closed extension of $k$, it amounts to the same to say that $A{\otimes }_{k}K$ is a finite separable $K$-algebra (4.6.1), so one sees that one can restrict to the case where $k$ is algebraically closed. Let us first show that $A$ is a finite $k$-algebra: it will suffice to show that every closed point $x$ of $X$ is isolated, since then the set of these points is open in $X$ and discrete, hence finite since it is quasi-compact ($X$ being Noetherian), which will establish our assertion by virtue of (I, 6.4.4). Now, one then has $k(x)=k$ since $k$ is algebraically closed (I, 6.4.3), hence there is a $Y$-section $s$ of $X$ such that $s(Y)=x$, and by virtue of (17.4.1.1), $x$ is the inverse image of the diagonal ${\Delta }_f(X)$ by a morphism $X\to X{\times }_{Y}X$, hence $x$ is open in $X$ by virtue of hypothesis b). One has thus shown that $A$ is a

finite $k$-algebra, direct composite of finite local $k$-algebras. To express that ${\Delta }_f$ is an open immersion, one can therefore restrict to the case where $A$ is a finite local $k$-algebra, $X=\mathrm{Spec}(A)$ being thus reduced to a single point; the residue field of $A$, being a finite extension of $k$, is necessarily identical to $k$, and consequently (I, 3.4.9) $X{\times }_{k}X$ is reduced to a single point and ${\Delta }_f$ is therefore necessarily an isomorphism. Now, since $A$ is a $k$-algebra, the canonical homomorphism $A{\otimes }_{k}A\to A$ can be bijective only if $A=k$. Q.E.D.

Remark (17.4.1.2). — Suppose only that $f$ is locally of finite type. Then ${\Omega }_{X/Y}^1$ is still an ${\mathcal{O}}_X$-Module of finite type (16.3.9), and ${\Delta }_f$ a morphism locally of finite presentation (1.4.3.1). The entire proof of (17.4.1) is then valid, provided that one replace a) by: the restriction of $f$ to a suitable neighbourhood of $x$ is a formally unramified morphism. One sees in addition that in this case the restriction of $f$ to a suitable neighbourhood of $x$ is a morphism locally quasi-finite.

Corollary (17.4.2).

Let $f:X\to Y$ be a morphism locally of finite presentation. The following properties are equivalent:

a) $f$ is unramified.

b) The diagonal morphism ${\Delta }_f:X\to X{\times }_{Y}X$ is an open immersion.

b') For every morphism $Y^{\prime }\to Y$, every $Y^{\prime }$-section of $X^{\prime }=X{\times }_Y{Y}^{\prime }$ is an open immersion.

c) One has ${\Omega }_{X/Y}^1=0$.

d) For every $y\in Y$, the $k(y)$-prescheme $f^{-1}(y)$ is unramified over $k(y)$.

d') For every $y\in Y$, the $k(y)$-prescheme $f^{-1}(y)$ is isomorphic to a prescheme of the form ${\coprod }_{\lambda \in L}\mathrm{Spec}(K_{\lambda })$, where, for each $\lambda \in L$, $K_{\lambda }$ is a finite separable extension of $k(y)$.

e) For every $x\in X$, the ring ${\mathcal{O}}_{X,x}$ is an ${\mathcal{O}}_{Y,f(x)}$-algebra formally unramified for the discrete topologies.

Corollary (17.4.3).

If $f:X\to Y$ is unramified, then $f$ is locally quasi-finite $(Er{r}_{III},20)$.

Proposition (17.4.4).

Let $Y$ be a locally Noetherian prescheme, $f:X\to Y$ a morphism locally of finite type, $x$ a point of $X$, $y=f(x)$. Set $A={\mathcal{O}}_{Y,y}$, $B={\mathcal{O}}_{X,x}$, which are Noetherian local rings, and let $k$ be the residue field of $A$. Then the equivalent conditions a) to e) of theorem (17.4.1) are also equivalent to each of the following:

f) $\hat{B}{\otimes }_{\hat{A}}k$ is a field, finite separable extension of $k$ (which implies that $\hat{B}$ is a finite Â-algebra).

f') $B$ is an $A$-algebra formally unramified for the adic topologies.

If moreover $k(x)=k(y)$, or if $k$ is separably closed, these conditions are also equivalent to:

f'') The homomorphism $\hat{A}\to \hat{B}$ is surjective.

Let us first note, by the same reasoning as in (0, 19.3.6), that it amounts to the same to say that $B$ is an $A$-algebra formally unramified for the preadic topologies, or that $\hat{B}$ is an Â-algebra formally unramified for the adic topologies. On the other hand, the hypothesis that $f$ is locally of finite type implies that ${\Omega }_{B/A}^1$

is a $B$-module of finite type (16.3.9), hence separated for the $\mathfrak{n}$-preadic topology (where $\mathfrak{n}$ is the maximal ideal of $B$) $(0_I,\mathrm{7.3.5})$; it amounts to the same to say that ${\Omega }_{B/A}^1=0$ or that ${\Omega }_{\hat{B}/\hat{A}}^1=0$; hence (0, 20.7.4), it amounts to the same to say that $B$ is an $A$-algebra formally unramified for the discrete topologies, or that $\hat{B}$ is an Â-algebra formally unramified for the preadic topologies. This proves the equivalence of conditions e) and f'). If $\mathfrak{m}$ is the maximal ideal of $A$, one has $k=A/\mathfrak{m}=\hat{A}/\mathfrak{m}\hat{A}$, so $\hat{B}{\otimes }_{\hat{A}}k=\hat{B}/\mathfrak{m}\hat{B}=\hat{B}{\otimes }_B(B/\mathfrak{m}B)$, and consequently $(0_I,\mathrm{7.3.5})$ $\hat{B}/\mathfrak{m}\hat{B}$ is the completion of $B/\mathfrak{m}B=B{\otimes }_{A}k$ for the $\mathfrak{n}$-preadic topology; this proves the equivalence of d'') and f). Finally, when $k(x)=k(y)$ or when $k$ is separably closed, the condition f) implies that the homomorphism $A/\mathfrak{m}A\to B/\mathfrak{n}B$ is bijective; the condition f) implies on the other hand that $B$ is a quasi-finite Â-algebra $(0_I,\mathrm{7.4.4})$, hence finite since Â is complete and $\hat{B}$ separated for the $\mathfrak{n}$-preadic topology, $\mathfrak{m}\hat{B}$ being an ideal of definition of $\hat{B}$ $(0_I,\mathrm{7.4.1})$. The homomorphism $\hat{A}\to \hat{B}$ is therefore surjective by virtue of Nakayama's lemma. Hence f) implies f''), and the converse is evident.

(17.4.5) Given an $S$-prescheme $Y$ and two $S$-morphisms $f:X\to Y$, $g:X\to Y$, one canonically deduces an $S$-morphism $(f,g{)}_S:X\to Y{\times }_{S}Y$. We shall call prescheme of coincidences of $f$ and $g$ the inverse image by $(f,g{)}_S$ of the diagonal ${\Delta }_{Y/S}$; it is therefore a sub-prescheme of $X$, which is closed when $Y$ is an $S$-scheme (I, 5.4.1).

Proposition (17.4.6).

Let $h:Y\to S$ be an unramified morphism and let $f:X\to Y$, $g:X\to Y$ be two $S$-morphisms. Then the prescheme $C$ of coincidences of $f$ and $g$ is a sub-prescheme induced on an open set of $X$; if moreover $Y$ is an $S$-scheme (I, 5.4.1), $C$ is a closed sub-prescheme of $X$.

Indeed, since ${\Delta }_{Y/S}:Y\to Y{\times }_{S}Y$ is an open immersion (17.4.2), the inverse image by $(f,g{)}_S$ of ${\Delta }_{Y/S}$ is a sub-prescheme induced on an open set of $X$ (I, 4.4.1). The last assertion results from (17.4.5).

Corollary (17.4.7).

Under the hypotheses of (17.4.6), let $x$ be a point of $X$ such that the two composite morphisms $\mathrm{Spec}(k(x))\to X{\to }^{f}Y$ and $\mathrm{Spec}(k(x))\to X{\to }^{g}Y$ are equal. Then there exists an open neighbourhood $U$ of $x$ such that $f|U=g|U$. If moreover $Y$ is an $S$-scheme, there exists an open and closed neighbourhood $X^{\prime }$ of $x$ in $X$ such that $f|X^{\prime }=g|X^{\prime }$. If finally one supposes in addition that $X$ is connected, one has $f=g$.

This results from (17.4.6) and (I, 5.3.17).

Corollary (17.4.8).

Under the hypotheses of (17.4.6), suppose that the structure morphism $\varphi =h\circ f=h\circ g$ from $X$ to $S$ is closed. Let $s$ be a point of $S$; let $X_s$ be the $k(s)$-prescheme ${\varphi }^{-1}(s)$ and suppose that the two composite morphisms $X_s\to X{\to }^{f}Y$ and $X_s\to X{\to }^{g}Y$ are equal. Then there exists an open neighbourhood $V$ of $s$ in $S$ such that $f|{\varphi }^{-1}(V)=g|{\varphi }^{-1}(V)$. If moreover $Y$ is an $S$-scheme and if $\varphi$ is open one can take $V$ open and closed. If finally one supposes in addition $S$ connected, one has $f=g$.

It results from (17.4.7) that the prescheme $C$ of coincidences of $f$ and $g$ is induced on an open of $X$ and contains $X_s$. Since $\varphi$ is closed, there exists an open neighbourhood $V$ of $s$ such that ${\varphi }^{-1}(V)\subset C$. If moreover $Y$ is an $S$-scheme, $C$ is closed, hence $\varphi (X-C)$ is at

once open and closed in $S$, and its complement $V$ in $S$ is therefore an open and closed neighbourhood of $s$ such that ${\varphi }^{-1}(V)\subset C$.

Proposition (17.4.9).

Let $Y$ be a connected prescheme, $f:X\to Y$ an unramified and separated morphism. Then every $Y$-section $g$ of $X$ is an isomorphism of $Y$ onto an open connected component of $X$, and the map $g\mapsto g(Y)$ is a bijection of $\Gamma (X/Y)$ onto the set of connected components $Z$ of $X$ (necessarily open in $X$) such that the restriction of $f$ to $Z$ be an isomorphism of $Z$ onto $Y$. In particular, if $g^{\prime }$ and g'' are two $Y$-sections of $X$ such that $g^{\prime }(y)=g^{\prime \prime }(y)$ for some $y\in Y$, one has $g^{\prime }=g^{\prime \prime }$.

It results indeed from (17.4.1, b'')) that a $Y$-section $s$ of $X$ is an open immersion, and since $X$ is a $Y$-scheme, $s$ is also a closed immersion (I, 5.4.7); it follows that $s$ is an isomorphism of $Y$ onto a sub-prescheme of $X$ induced on an open and closed part of $X$, and since $s(Y)$ is connected, it is necessarily a connected component of $X$. The rest of the proposition is immediate.

Remark (17.4.10). — Taking the remark (17.4.1.2) into account, one sees that, in the statements (17.4.6) to (17.4.9), one can everywhere replace the words "unramified" by "formally unramified and locally of finite type".

17.5. Characterizations of smooth morphisms

Theorem (17.5.1).

Let $f:X\to Y$ be a morphism locally of finite presentation, $x$ a point of $X$, $y=f(x)$. The following conditions are equivalent:

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

b) $f$ is flat at the point $x$ and the $k(y)$-prescheme $f^{-1}(y)$ is smooth over $k(y)$ at the point $x$.

b') $f$ is regular at the point $x$ (6.8.1).

c) The ring ${\mathcal{O}}_{X,x}$ is an ${\mathcal{O}}_{Y,y}$-algebra formally smooth for the discrete topologies.

One can restrict to the case where $Y=\mathrm{Spec}(A)$, $X=\mathrm{Spec}(C)$, where $C=B/\mathfrak{j}$, $B=A[T_1,\cdots ,T_n]$ being a polynomial algebra and $\mathfrak{j}$ an ideal of $B$ of finite type. The equivalence of a) and c) then results from the equivalence of a) and c) in (0, 22.6.4). On the other hand, applying this result to the morphism locally of finite type $f^{-1}(y)\to \mathrm{Spec}(k(y))$, one sees that the equivalence of b) and b') results from the equivalence of a) and b) in (6.8.6). It thus remains to prove the equivalence of a) and b).

Let us first show that a) implies b); denote by $\mathfrak{p}$ the prime ideal ${\mathfrak{j}}_x$ in $C$, $\mathfrak{r}$ the prime ideal ${\mathfrak{j}}_y$ in $A$; one has $\mathfrak{p}=\mathfrak{q}/\mathfrak{j}$, where $\mathfrak{q}$ is a prime ideal of $B$ and $\mathfrak{r}$ is the inverse image of $\mathfrak{q}$ in $A$. The hypothesis a) implies first that $f^{-1}(y)$ is smooth over $k(y)$ at the point $x$ by (17.3.3), and it is a question of showing moreover that $C_{\mathfrak{p}}={\mathcal{O}}_{X,x}$ is a flat $A_{\mathfrak{r}}$-module. Since $C_{\mathfrak{p}}$ is a formally smooth $A_{\mathfrak{r}}$-algebra and $B$ is a formally smooth $A$-algebra (for the discrete topologies), the Jacobian criterion (0, 22.6.4), together with (0, 19.1.12), implies that there exists in $\mathfrak{j}$ a system of $r$ polynomials $g_i$ $(1⩽i⩽r)$ and $r$ indices $j_i$ $(1⩽i⩽r)$ such that the images of the $g_i$ in ${\mathfrak{j}}_{\mathfrak{q}}/{\mathfrak{j}}_{\mathfrak{q}}^2$ generate this $B_{\mathfrak{q}}$-module and that one has

$$(\mathrm{17.5.1.1})det(\partial g_i/\partial T_{j_k})\notin \mathfrak{q}.$$

Let us now note that if $g:\mathrm{Spec}(B)\to \mathrm{Spec}(A)$ is the structure morphism, the fibre $g^{-1}(y)$ is the spectrum of the regular ring $k(y)[T_1,\cdots ,T_n]$ (0, 17.3.7), hence the Noetherian local ring $B_{\mathfrak{q}}/\mathfrak{r}{B}_{\mathfrak{q}}$ at a point of this fibre is regular. Now, condition (17.5.1.1) implies that the canonical images ${\bar{u}}_i$ of the $g_i$ in the maximal ideal $\mathfrak{m}$ of $B_{\mathfrak{q}}/\mathfrak{r}{B}_{\mathfrak{q}}$ are linearly independent mod. ${\mathfrak{m}}^2$: otherwise, there would exist polynomials $w_i\in B$ $(1⩽i⩽r)$ not all belonging to $\mathfrak{q}$ and such that ${\sum }_{i=1}^r{w}_i{g}_i\in {\mathfrak{q}}^2$. Differentiating with respect to the $T_{j_k}$, one would conclude that $\sum w_i(\partial g_i/\partial T_{j_k})\in \mathfrak{q}$ for $1⩽k⩽r$, which would contradict (17.5.1.1) since $\mathfrak{q}$ is prime. One concludes therefore from (0, 17.1.7) that $({\bar{u}}_i)$ is a regular sequence in $B_{\mathfrak{q}}/\mathfrak{r}{B}_{\mathfrak{q}}$. But since the morphism $g$ is locally of finite presentation and $B$ is a flat $A$-module, it results from (11.3.8) that the canonical images $u_i^{\prime }$ of the $g_i$ in $B_{\mathfrak{q}}$ also form a regular sequence and that $B_{\mathfrak{q}}/(\sum u_i^{\prime }{B}_{\mathfrak{q}})$ is a flat $A_{\mathfrak{r}}$-module. Since the images of the $u_i^{\prime }$ in ${\mathfrak{j}}_{\mathfrak{q}}/{\mathfrak{j}}_{\mathfrak{q}}^2$ generate this $B_{\mathfrak{q}}$-module, it results from Nakayama's lemma that one has $\sum u_i^{\prime }{B}_{\mathfrak{q}}={\mathfrak{j}}_{\mathfrak{q}}$, and $C_{\mathfrak{p}}=B_{\mathfrak{q}}/{\mathfrak{j}}_{\mathfrak{q}}$ is therefore indeed a flat $A_{\mathfrak{r}}$-module.

Let us finally prove that b) implies a). With the same notation, the hypothesis that $B_{\mathfrak{q}}/{\mathfrak{j}}_{\mathfrak{q}}$ is a flat $A_{\mathfrak{r}}$-module implies that the canonical homomorphism ${\mathfrak{j}}_{\mathfrak{q}}/\mathfrak{r}{\mathfrak{j}}_{\mathfrak{q}}\to B_{\mathfrak{q}}/\mathfrak{r}{B}_{\mathfrak{q}}$ is injective $(0_I,\mathrm{6.1.2})$, so that ${\mathfrak{j}}_{\mathfrak{q}}/\mathfrak{r}{\mathfrak{j}}_{\mathfrak{q}}$ is identified with an ideal of $B_{\mathfrak{q}}/\mathfrak{r}{B}_{\mathfrak{q}}$. Since $B_{\mathfrak{q}}/\mathfrak{r}{B}_{\mathfrak{q}}$ is an $A_{\mathfrak{r}}$-algebra formally smooth for the discrete topologies, one can apply, to $C_{\mathfrak{p}}=(B_{\mathfrak{q}}/\mathfrak{r}{B}_{\mathfrak{q}})/({\mathfrak{j}}_{\mathfrak{q}}/\mathfrak{r}{\mathfrak{j}}_{\mathfrak{q}})$, the Jacobian criterion (0, 22.6.4); together with (0, 19.1.12), the latter proves, by virtue of hypothesis b), the existence of $r$ polynomials $v_i\in k(y)[T_1,\cdots ,T_n]$ such that their images in $({\mathfrak{j}}_{\mathfrak{q}}/\mathfrak{r}{\mathfrak{j}}_{\mathfrak{q}})/({\mathfrak{j}}_{\mathfrak{q}}/\mathfrak{r}{\mathfrak{j}}_{\mathfrak{q}}{)}^2$ generate this $(B_{\mathfrak{q}}/\mathfrak{r}{B}_{\mathfrak{q}})$-module and that one has

  (17.5.1.2)    det(∂v_i/∂T_{j_k}) ∉ 𝔮 B_𝔮/𝔯 B_𝔮.

If, for each $i$, one then designates by $g_i$ an element of $\mathfrak{j}$ whose $v_i$ is the canonical image, it follows from (17.5.1.2) that the $g_i$ verify condition (17.5.1.1); on the other hand, by virtue of Nakayama's lemma, the images $u_i^{\prime }$ of the $g_i$ in ${\mathfrak{j}}_{\mathfrak{q}}$ generate this $B_{\mathfrak{q}}$-module. The Jacobian criterion (0, 22.6.4) together with (0, 19.1.12) then proves that $C_{\mathfrak{p}}=B_{\mathfrak{q}}/{\mathfrak{j}}_{\mathfrak{q}}$ is an $A_{\mathfrak{r}}$-algebra formally smooth for the discrete topologies. Q.E.D.

Corollary (17.5.2).

Let $f:X\to Y$ be a morphism locally of finite presentation. For $f$ to be smooth (in the sense of (17.3.1)), it is necessary and sufficient that $f$ be regular (6.8.1), in other words that $f$ be flat and that for every $y\in Y$, $f^{-1}(y)$ be a geometrically regular $k(y)$-prescheme (6.7.6).

One has thus established the equivalence of the two definitions of "smooth morphism" given in (6.8.1) and (17.3.1).

Proposition (17.5.3).

Let $Y$ be a locally Noetherian prescheme, $f:X\to Y$ a morphism locally of finite type, $x$ a point of $X$, $y=f(x)$. Set $A={\mathcal{O}}_{Y,y}$, $B={\mathcal{O}}_{X,x}$, which are Noetherian local rings. Then the equivalent conditions a) to c) of (17.5.1) are also equivalent to each of the following:

d) $B$ is an $A$-algebra formally smooth for the preadic topologies.

d') $\hat{B}$ is an Â-algebra formally smooth for the adic topologies.

If moreover $k(x)=k(y)$, these conditions are also equivalent to:

d'') $\hat{B}$ is an Â-algebra isomorphic to a formal power series algebra $\hat{A}[[T_1,\cdots ,T_n]]$.

The equivalence of condition c) of (17.5.1) and d) results from the equivalence of a) and d) in the Jacobian criterion (0, 22.6.4), and the equivalence of d) and d') results from (0, 19.3.6). On the other hand, d'') implies d') without any hypothesis on the residue fields (0, 19.3.4). Finally, if $\mathfrak{m}$ designates the maximal ideal of Â, the hypothesis d') implies that $\hat{B}/\mathfrak{m}\hat{B}$ is a complete Noetherian local $k(y)$-algebra, formally smooth for its adic topology (0, 19.3.5); the hypothesis $k(y)=k(x)$ then implies that $\hat{B}/\mathfrak{m}\hat{B}$ is $k(y)$-isomorphic to a formal power series algebra $k(y)[[T_1,\cdots ,T_n]]$ (0, 19.6.4). Since on the other hand, $\hat{A}[[T_1,\cdots ,T_n]]$ is a flat Â-module and a complete Noetherian local Â-algebra, one concludes from (0, 19.7.1.5) that this algebra is isomorphic to $\hat{B}$. Hence d') implies d'') under the additional hypothesis $k(x)=k(y)$.

Remark (17.5.4). — Suppose that $Y$ is a locally Noetherian prescheme, and $f:X\to Y$ a morphism locally of finite type. The criterion (17.5.3, d), together with (0, 22.1.4), shows that to prove that $f$ is smooth, one can apply definition (17.1.1), restricting to the case where the affine scheme $Y^{\prime }$ is the spectrum of an Artinian local ring.

Proposition (17.5.5).

Let $Y$ be a locally Noetherian prescheme, $f:X\to Y$ a morphism locally of finite type, $x$ a point of $X$, $y=f(x)$. Suppose that $Y$ is reduced at the point $y$. Then, for $f$ to be smooth at the point $x$, it is necessary and sufficient that $f$ be universally open in a neighbourhood of $x$ in $f^{-1}(y)$ and that $f^{-1}(y)$ be a geometrically regular $k(y)$-prescheme at the point $x$.

Taking (17.5.1) into account, everything reduces to seeing that, if $f^{-1}(y)$ is a geometrically regular $k(y)$-prescheme at the point $x$, it is equivalent to say that $f$ is flat at the point $x$ or universally open in a neighbourhood of $x$ in $f^{-1}(y)$. Now, if $f$ is flat at the point $x$, it is so in a neighbourhood of $x$ in $X$ (11.1.1) and consequently universally open in this neighbourhood (2.4.6). Conversely, the hypothesis that $f$ is universally open in a neighbourhood of $x$ in $f^{-1}(y)$ and that $f^{-1}(y)$ is a geometrically regular $k(y)$-prescheme at the point $x$ implies that $f$ is flat at the point $x$, since ${\mathcal{O}}_{Y,y}$ is reduced (15.2.2).

Corollary (17.5.6).

Let $Y$ be a locally Noetherian prescheme, $f:X\to Y$ a morphism locally of finite type, $x$ a point of $X$, $y=f(x)$. Suppose that $Y$ is reduced and geometrically unibranch (6.15.1) at the point $y$. Then, for $f$ to be smooth at the point $x$, it is necessary and sufficient that $f$ be equidimensional at the point $x$ and that $f^{-1}(y)$ be a geometrically regular $k(y)$-prescheme at the point $x$.

Noting that the set of points where $f$ is equidimensional is open (13.3.2), one sees that the corollary results from (17.5.5) and from Chevalley's criterion (14.4.4).

The fact that $f$ is a smooth morphism at a point implies in particular that $f$ verifies at this point

all the properties defined in (6.8.1). One has therefore the following properties, which we recall for the convenience of references:

Proposition (17.5.7).

Let $f:X\to Y$ be a morphism locally of finite presentation, smooth at a point $x\in X$; set $y=f(x)$. Then, for the ring ${\mathcal{O}}_{X,x}$ to be reduced (resp. integrally closed, resp. geometrically unibranch), it is necessary and sufficient that ${\mathcal{O}}_{Y,y}$ be so.

This has indeed been proved in (11.3.13) and (11.3.14), completed by $Er{r}_{IV},30$.

Proposition (17.5.8).

Let $Y$ be a locally Noetherian prescheme, $f:X\to Y$ a morphism locally of finite type, smooth at a point $x\in X$; set $y=f(x)$. Then:

(i) One has dim(𝒪_{X, x}) = dim(𝒪_{Y, y}) + dim(𝒪_{X, x} ⊗_{𝒪_{Y, y}} k(y)).

(ii) One has $coprof({\mathcal{O}}_{X,x})=coprof({\mathcal{O}}_{Y,y})$.

(iii) For the ring ${\mathcal{O}}_{X,x}$ to possess property $(S_n)$ (5.7.2) (resp. $(R_n)$ (5.8.2)), it is necessary and sufficient that the ring ${\mathcal{O}}_{Y,y}$ possess it. In particular, for ${\mathcal{O}}_{X,x}$ to be regular, it is necessary and sufficient that ${\mathcal{O}}_{Y,y}$ be so.

These are particular cases of (6.1.2), (6.3.2), (6.4.1), and (6.5.3).

17.6. Characterizations of étale morphisms

Theorem (17.6.1).

Let $f:X\to Y$ be a morphism locally of finite presentation, $x$ a point of $X$, $y=f(x)$. The following conditions are equivalent:

a) $f$ is étale at the point $x$.

a') $f$ is smooth at the point $x$ and unramified at the point $x$.

b) $f$ is smooth at the point $x$ and quasi-finite at the point $x$ $(Er{r}_{III},20)$.

c) $f$ is flat at the point $x$ and unramified at the point $x$.

c') $f$ is flat at the point $x$ and the ring ${\mathcal{O}}_{X,x}/{\mathfrak{m}}_y{\mathcal{O}}_{X,x}$ is a field, finite separable extension of $k(y)$.

d) The ring ${\mathcal{O}}_{X,x}$ is an ${\mathcal{O}}_{Y,y}$-algebra formally étale for the discrete topologies.

The equivalence of a) and a') results at once from the definitions; that of a) and d) results from the equivalence of a) and e) in (17.4.1) and of the equivalence of a) and c) in (17.5.1). The equivalence of c) and c') results from the equivalence of a) and d'') in (17.4.1). The fact that a') implies c') follows from (17.5.1); conversely, if c') is verified, $f$ is regular (hence smooth by (17.5.1)) at the point $x$, since if $K$ is a finite separable extension of a field $k$, then, for every extension $k^{\prime }$ of $k$, $\mathrm{Spec}(K{\otimes }_k{k}^{\prime })$ is regular, being the sum of a finite number of spectra of fields. The fact that a') implies b) results from (17.4.1, d') and (17.5.1, b). It remains therefore to see that b) implies c), and since one already knows that $f$ is flat at the point $x$ by (17.5.1), it suffices to show that $f^{-1}(y)$ is a $k(y)$-prescheme unramified over $k(y)$; in other words, one is reduced to proving that b) implies c) when $Y=\mathrm{Spec}(k)$ is the spectrum of a field $k$. Since the question is local on $X$, one can restrict to the case where $X=\mathrm{Spec}(A)$, where $A$ is a finite local $k$-algebra $(0_I,\mathrm{7.4.1})$. By virtue of hypothesis b), $A$ is a $k$-algebra formally smooth for the discrete topologies, which coincide here with the preadic topologies; hence (0, 19.6.5), $A$ is a regular local ring, hence

a field since it is Artinian, and it then results from (0, 19.6.5.1) that $A$ must be a finite separable extension of $k$, which completes the proof (17.4.1).

Corollary (17.6.2).

Let $f:X\to Y$ be a morphism locally of finite presentation. The following conditions are equivalent:

a) $f$ is étale.

a') $f$ is smooth and unramified.

b) $f$ is smooth and locally quasi-finite $(Er{r}_{III},20)$.

c) $f$ is flat and unramified.

c') $f$ is flat, and every fibre $f^{-1}(y)$ is a $k(y)$-scheme sum of spectra of fields, finite separable extensions of $k(y)$.

c'') $f$ is flat, and for every $y\in Y$ and every algebraically closed extension $k^{\prime }$ of $k(y)$, the "geometric fibre" $f^{-1}(y){\otimes }_{k(y)}{k}^{\prime }$ is a sum of spectra of fields isomorphic to $k^{\prime }$.

The only point that remains to prove is the equivalence of c') and c''). It is clear that c') implies c'') by base change (17.3.3). On the other hand, since the projection morphism $f^{-1}(y){\otimes }_{k(y)}{k}^{\prime }\to f^{-1}(y)$ is open (2.4.10), the hypothesis c'') implies that the space $f^{-1}(y)$ is discrete, hence, for every point $x\in X_y=f^{-1}(y)$, the local ring ${\mathcal{O}}_{X_y,x}=A$ is a finite $k(y)$-algebra, hence an Artinian local ring; in addition it results from c'') that $\mathrm{Spec}(A{\otimes }_{k(y)}{k}^{\prime })$ is a sum of spectra of fields isomorphic to $k^{\prime }$, which is possible only if $A$ is a field, finite separable extension of $k(y)$ (4.6.1).

Proposition (17.6.3).

Let $Y$ be a locally Noetherian prescheme, $f:X\to Y$ a morphism locally of finite type, $x$ a point of $X$, $y=f(x)$. Set $A={\mathcal{O}}_{Y,y}$, $B={\mathcal{O}}_{X,x}$, which are Noetherian local rings, and let $k$ be the residue field of $A$. Then the equivalent conditions a) to d) of (17.6.1) are also equivalent to each of the following:

e) $B$ is an $A$-algebra formally étale for the adic topologies.

e') $\hat{B}$ is a free Â-module and $\hat{B}{\otimes }_{\hat{A}}k$ is a field, finite separable extension of $k$ (which implies that $\hat{B}$ is a finite Â-algebra).

If moreover $k(x)=k(y)$, or if $k$ is separably closed, these conditions are also equivalent to:

e'') The canonical homomorphism $\hat{A}\to \hat{B}$ is bijective.

The equivalence of e) with each of the conditions of (17.6.1) results at once from (17.4.4, f') and (17.5.3, d'). The fact that e) implies e') results from (17.4.4, f) and from (0, 19.7.1), taking into account that $\hat{B}$ is then a finite Â-algebra (17.4.4) and that it amounts to the same to say that $\hat{B}$ is a flat Â-module or a free Â-module $(0_{III},\mathrm{10.1.3})$. Conversely, the fact that e') implies e) results from (17.4.4) and from (0, 19.7.1). Finally, e') implies that the homomorphism $\hat{A}\to \hat{B}$ is injective, and if $k(x)=k(y)$ or if $k$ is separably closed, this homomorphism is surjective by (17.4.4). The converse is immediate.

Proposition (17.6.4).

Under the hypotheses of (17.6.3), if $f$ is étale at the point $x$, one has $\dim ({\mathcal{O}}_{X,x})=\dim ({\mathcal{O}}_{Y,y})$.

This is a particular case of (17.5.8, (i)) since $x$ is isolated in its fibre $f^{-1}(y)$.

17.7. Descent properties, passage to the limit, and constructibility

Proposition (17.7.1).

Let $f:X\to Y$ be a morphism locally of finite presentation, $g:Y^{\prime }\to Y$ a morphism, $X^{\prime }=X{\times }_Y{Y}^{\prime }$, $f^{\prime }=f_{(Y^{\prime })}:X^{\prime }\to Y^{\prime }$ and $g^{\prime }:X^{\prime }\to X$ the canonical projections. Let $x^{\prime }$ be a point of $X^{\prime }$ and set $x=g^{\prime }(x^{\prime })$, $y^{\prime }=f^{\prime }(x^{\prime })$.

(i) If $f^{\prime }$ is unramified at the point $x^{\prime }$, then $f$ is unramified at the point $x$.

(ii) Suppose moreover that $g$ is flat at the point $y^{\prime }$. Then, if $f^{\prime }$ is smooth (resp. étale) at the point $x^{\prime }$, $f$ is smooth (resp. étale) at the point $x$.

Set $y=f(x)=g(y^{\prime })$, so that one has $f^{\prime -1}(y^{\prime })=f^{-1}(y){\otimes }_{k(y)}k(y^{\prime })$; note that $f^{\prime }$ is locally of finite presentation.

(i) Since the property for a morphism (locally of finite presentation) of being unramified at a point involves only the fibre of the morphism at that point (17.4.1, d), one can restrict to the case where $Y$ and $Y^{\prime }$ are spectra of fields. But then it amounts to the same to say that $f$ (resp. $f^{\prime }$) is unramified at $x$ (resp. $x^{\prime }$) or that it is étale at this point (17.6.1, c), hence (i) is a consequence of (ii).

(ii) Since $f^{\prime }$ is flat at the point $x^{\prime }$ (17.5.1), the hypothesis that $g$ is flat at the point $y^{\prime }$ implies that $f$ is flat at the point $x$, since the projection $g^{\prime }:X^{\prime }\to X$ is a morphism flat at the point $x^{\prime }$, and $f\circ g^{\prime }=g\circ f^{\prime }$ is a morphism flat at the point $x^{\prime }$, whence the conclusion (2.2.11, (iv)). The fact that $f$ is smooth (resp. étale) at the point $x$ then involves only the fibre $f^{-1}(y)$ (17.5.1 and 17.6.1), and one is therefore again reduced to the case where $Y=\mathrm{Spec}(k)$ and $Y^{\prime }=\mathrm{Spec}(k^{\prime })$ are spectra of fields. To say that $f^{\prime }$ is smooth at the point $x^{\prime }$ then signifies (17.5.1) that $X^{\prime }$ is a geometrically regular $k^{\prime }$-prescheme at the point $x^{\prime }$, and this implies (6.7.8) that $X$ is a geometrically regular $k$-prescheme at the point $x$, hence that $f$ is smooth at the point $x$. Suppose in addition that $f^{\prime }$ is étale at the point $x^{\prime }$, so that $x^{\prime }$ is isolated in $f^{\prime -1}(y^{\prime })$ (17.6.1); since the projection $f^{\prime -1}(y^{\prime })\to f^{-1}(y)$ is an open morphism (2.4.10), $x$ is isolated in $f^{-1}(y)$; since one already knows that $f$ is smooth at the point $x$, it is étale at this point (17.6.1).