Proposition 23

Let A be an integral K-algebra, B its field of fractions and u: A → B the canonical injection. Then Ω_0(u): Ω_K(A) ⊗_A B → Ω_K(B) is an isomorphism.

Using the fact that in diagram (42) the vertical arrows are bijective, it reduces to proving that, for every vector space E over B, the mapping C(u): D_K(B, E) → D_K(A, E) is bijective. But this follows from the fact that every K-derivation of A into E can be extended uniquely to a K-derivation of B into E (no. 5, Proposition 5).