Please let us know about mistakes in these notes! 1. Module of Differentials

Definition 1.2. The module of relative differential forms of B over A is a Bmodule ΩB/A, together with an A-derivation d : B → ΩB/A satisfying the following universal property: for any B-module M and for any A-derivation d : B → M , there exists a unique B-module homomorphism f : ΩB/A → M such that d ′ = f ◦d. If DerA(B,M) denotes the set of all A-derivations from B into M , then we have a natural bijection DerA(B,M) ↔ HomB(ΩB/A,M). Proposition 1.3. The module of relative differential forms ΩB/A exists and is unique up to a unique isomorphism of B-modules.