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Commutative algebra

Yoshifumi Tsuchimoto

\fbox{04. Derivations and differentials}

LEMMA 04.1   Let $ A$ be a commutative ring. Then for the polynomial ring $ A=A[X_1,X_2,\dots,X_n]$ of $ n$ -variables over $ A$ , the module $ \Omega^1_{B/A}$ of $ 1$ -differentials of $ B$ over $ A$ is equal to a free module generated by $ d X_1 , d X_2,\dots d X_n$ . Namely, we have

$\displaystyle \Omega^1_{B/A}=A d X_1 \oplus A d X_2 \oplus\dots \oplus A d X_n.
$

LEMMA 04.2   Let $ k$ be a ring. Let $ A,B$ be $ k$ -algebras. Then for any $ k$ -algebra homomorphism $ \varphi:A\to B$ we have

$\displaystyle B\otimes_A \Omega^1_{A/k} \to \Omega^1_{B/k}\to \Omega^1_{B/A} \to 0
$

LEMMA 04.3   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:

$\displaystyle I/I^2\to \Omega^1_{B/A}/I \Omega^1_{B/A}\to \Omega^1_{(B/I)/A} \to 0
$

where the first arrow maps $ f \mod I^2$ to $ df $ .



2011-05-26