COMMUTATIVE ALGEBRA

YOSHIFUMI TSUCHIMOTO

03. Derivations and differentials

Definition 3.1. 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 3.2. Let A be a commutative ring. Let B be an Aalgebra. Let M be an B-module. A map D : A → M is called an A-derivation if it satisfies the following conditions.

(1) D is A-linear.

(2) D(xy) = xD(y) + D(x)y (∀x, ∀y ∈ B).

Proposition 3.3. For any algebra B over a ring A, There exists an universal derivation d : B → Ω_{B/A}^{1}.

Definition 3.4. The module Ω_{B/A}^{1} is called the module of differentials of B over A.

Definition 3.5. An algebra B over a ring A is called unramified over A if Ω_{B/A}^{1} = 0. B is called étale over A if it is unramified and flat.

Lemma 3.6. Let A be a commutative ring. Let S be its multiplicative subset. Then S^{−1}A is étale over A.