COMMUTATIVE ALGEBRA

YOSHIFUMI TSUCHIMOTO

04. Derivations and differentials

Lemma 4.1. Let A be a commutative ring. Then for the polynomial ring A = A[X_{1}, X_{2}, …, X_{n}] of n-variables over A, the module Ω_{B/A}^{1} of 1-differentials of B over A is equal to a free module generated by dX_{1}, dX_{2}, … dX_{n}. Namely, we have

Ω_{B/A}^{1} = AdX_{1} ⊕ AdX_{2} ⊕ ⋯ ⊕ AdX_{n}.

Lemma 4.2. Let k be a ring. Let A, B be k-algebras. Then for any k-algebra homomorphism φ : A → B we have

B ⊗_{A} Ω_{A/k}^{1} → Ω_{B/k}^{1} → Ω_{B/A}^{1} → 0

Lemma 4.3. Let A be a commutative ring. Let B be a commutative A-algebra. Then for any ideal I of B, we have the following exact sequence:

I/I^{2} → Ω_{B/A}^{1}/IΩ_{B/A}^{1} → Ω_{(B/I)/A}^{1} → 0

where the first arrow maps f mod I^{2} to df.