II. Smooth morphisms: generalities, differential properties

References to Exp. I are indicated by an “I”. We recall that all rings are assumed to be Noetherian, and all preschemes locally Noetherian.

1. Generalities

Let Y be a prescheme, and t_1,\ldots,t_n indeterminates; we set

Y[t_1,\ldots,t_n] = Y\otimes_\mathbb{Z}\mathbb{Z}[t_1,\ldots,t_n]. \tag{1.1}

Then Y[t_1,\ldots,t_n] is an affine Y-scheme, defined by the quasi-coherent sheaf of algebras {\mathcal{O}}_Y[t_1,\ldots,t_n]. The data of a section of this prescheme over Y is thus equivalent to the data of n sections of {\mathcal{O}}_Y (corresponding to the images of the t_i under the corresponding homomorphism). If Y' is over Y, then we have

Y[t_1,\ldots,t_n] \times_Y Y' = Y'[t_1,\ldots,t_n] \tag{1.2}

(which implies that the data of a Y-morphism from Y' to Y[t_1,\ldots,t_n] is equivalent to the data of n sections of {\mathcal{O}}_{Y'}). We also have

\big(Y[t_1,\ldots,t_n]\big)[t_{n+1},\ldots,t_m] = Y[t_1,\ldots,t_m] \tag{1.3}

by the analogous formula for rings of polynomials over \mathbb{Z}. Equation (1.2) implies that Y[t_1,\ldots,t_n] varies functorially with respect to Y.

Further, Y[t_1,\ldots,t_n] is flat and of finite type over Y.

Let f\colon X\to Y be a morphism, making X a Y-prescheme. We say that f is smooth14 at x\in X, or that X is smooth over Y at x, if there exists an integer n\geqslant 0, an open neighbourhood U of x, and an étale Y-morphism from U to Y[t_1,\ldots,t_n]. We say that f (resp. X) is smooth if it is smooth at all points of X. An algebra B over a ring A is said to be smooth at a prime ideal {\mathfrak{p}} of B if \operatorname{Spec}(B) is smooth over \operatorname{Spec}(A) at the point {\mathfrak{p}}; we say that B is smooth over A if it smooth smooth over A at all prime ideals {\mathfrak{p}} of B. Finally, a local homomorphism A\to B of local rings is said to be smooth, or B is said to be smooth15 over A, if B is the localisation of an algebra of finite type B_1 that is smooth over A.

We note that the notion of smoothness of X over Y is local in X and in Y; if X is smooth over Y, then it is locally of finite type over Y.

The set of points x of X at which f is smooth is open.

Proof. This is trivial, by definition.

If B is smooth over A at {\mathfrak{p}}, then it is smooth over A at {\mathfrak{q}} for every prime ideal {\mathfrak{q}} of B that is contained in {\mathfrak{p}}.

Note that 1.1 also implies that the last two definitions of Definition 1.1 agree on their common domain of existence.

  1. An étale morphism (in particular, an open immersion, or an identity morphism) is smooth.
  2. A base extension of a smooth morphism is a smooth morphism.
  3. The composition of two smooth morphisms is smooth.

Proof.

  1. This is trivial by definition; more precisely, see 1.4.
  2. This follows immediately from the analogous fact for étale morphisms (I 4.6) and for the projections Y[t_1,\ldots,t_n]\to Y (cf. (1.2)).
  3. This follows formally from the fact that it is separately true for both “étale” (I 4.6) and for projections of the type Y[t_1,\ldots,t_n]\to Y (cf. (1.3)), and from the two facts cited for (ii): Suppose that Y is smooth over Z, and that X is smooth over Y, then we can prove that X is smooth over Z; we can suppose Y to be étale over Z[t_1,\ldots,t_n], and X to be étale over Y[s_1,\ldots,s_m], then the first hypothesis implies that Y[s_1,\ldots,s_m] is étale over Z[t_1,\ldots,t_n][s_1,\ldots,s_m]=Z[t_1,\ldots,s_m], and so X is étale over Z[t_1,\ldots,s_m], qed.

Étale = quasi-finite + smooth.

The integer n which appears in Definition 1.1 is well defined, since we immediately see that it is the dimension of the local ring of x in its fibre f^{-1}(f(x)). We call this the relative dimension of X over Y. It behaves additively under composition of morphisms.

2. Some smoothness criteria for morphisms


  1. Old terminology: f is simple at x, or x is a simple point for f. This terminology led to confusion in many contexts (simple algebras, simple groups) and had to be abandoned.↩︎

  2. It is better to say, as in EGA IV 18.6.1, that B is essentially smooth over A.↩︎