Commutative algebra

Yoshifumi Tsuchimoto

DEFINITION 03.1   Let $A$ be a ring. A ring $B$ is called an $A$ -algebra if there is given a distinguished ring homomorphism (which is called the structure morphism) from $A$ to $B$ .

DEFINITION 03.2   Let $A$ be a commutative ring. Let $B$ be an $A$ -algebra. Let $M$ be an $B$ -module. A map $D:A\to M$ is called an $A$ -derivation if it satisfies the following conditions.

1. $D$ is $A$ -linear.
2. $D(x y)=x D(y)+ D(x) y \qquad (\forall x ,\forall y \in B)$ .

PROPOSITION 03.3   For any algebra $B$ over a ring $A$ , There exists an universal derivation $d:B\to \Omega^1_{B/A}$ .

DEFINITION 03.4   The module $\Omega^1_{B/A}$ is called the module of differentials of $B$ over $A$ .

DEFINITION 03.5   An algebra $B$ over a ring $A$ is called unramified over $A$ if $\Omega^1_{B/A}=0$ . $B$ is called étale over $A$ if it is unramified and flat.

LEMMA 03.6   Let $A$ be a commutative ring. Let $S$ be its multiplicative subset. Then $S^{-1}A$ is étale over $A$ .