## 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 *smooth* 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 smooth 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.

—

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

*Proof*. —

- This is trivial by definition; more precisely, see 1.4.
- 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)).
- 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.