Commutative algebra

Yoshifumi Tsuchimoto

Let us recall the universality of polynomial algebras.

PROPOSITION 08.1   Let $A$ be a commutative ring. Let $B$ be an $A$ -algebra. That means, we assume that there is given a specific homomorphism (called the structure homomorphism) $\iota: A\to B$ . Then for any family $\{b_\lambda\}_{\lambda \in \Lambda}$ of elements of $B$ , there exists a unique ring homomorphism

$$\varphi:A[\{X_\lambda\}_{\lambda \in \Lambda}] \to B$$

such that $\varphi\vert _A=\iota$ and $\varphi(X_\lambda)=b_\lambda$ for all $\lambda \in \Lambda$ .

As a corollary, we see:

PROPOSITION 08.2   Let $A$ be a commutative ring. Then any polynomial algebra $A[\{X_\lambda\}_{\lambda \in \Lambda}$ is 0-smooth over $A$ .

LEMMA 08.3   Let $A$ be a ring. Let $B$ be an $A$ -algebra. Let $I$ be a finitely generated ideal of $B$ . Let us denote by $\hat B=\varprojlim (B/I^n)$ (respectively, $\hat I$ ) the completion of $B$ (respectively, $I$ ) with respect to the $I$ -adic topology. Then $B$ is $I$ -smooth over $A$ if and only if $\hat B$ is $\hat I$ -smooth over $A$ .

PROOF.. $B/I^n \cong \hat B/ \hat I^n$ for any $n$ .

$\qedsymbol$

COROLLARY 08.4   Let $A$ be a ring. Then $A[[X_1,X_2,\dots, X_n]]$ is $I$ -smooth over $A$ for $I=\sum_{i=1}^n X_i A[[X_1,X_2,\dots, X_n]]$ .

Note. In general, $A[[X]]$ is not 0 -smooth over $A$ . See [1] and the literatures cited there.

THEOREM 08.5   Let $A$ be a ring. Let $B$ be an $A$ -algebra with an ideal $I$ . If $B/I$ is 0 -smooth over $A$ , then the sequence (which appears in Lemma 04.2)

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

is split exact.

The following theorem says that the converse is true if the ring $B$ is 0 -smooth.

THEOREM 08.6   Let $A$ be a ring. Let $B$ be an $A$ -algebra with an ideal $I$ . Assume $B$ is 0 -smooth over $A$ . If the exact sequence

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

is split exact, then $B/I$ is 0 -smooth over $A$ .