Next: Bibliography

# Commutative algebra

Yoshifumi Tsuchimoto

Let us recall the universality of polynomial algebras.

PROPOSITION 08.1   Let be a commutative ring. Let be an -algebra. That means, we assume that there is given a specific homomorphism (called the structure homomorphism) . Then for any family of elements of , there exists a unique ring homomorphism

such that and for all .

As a corollary, we see:

PROPOSITION 08.2   Let be a commutative ring. Then any polynomial algebra is 0-smooth over .

LEMMA 08.3   Let be a ring. Let be an -algebra. Let be a finitely generated ideal of . Let us denote by (respectively, ) the completion of (respectively, ) with respect to the -adic topology. Then is -smooth over if and only if is -smooth over .

PROOF.. for any .

COROLLARY 08.4   Let be a ring. Then is -smooth over for .

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

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

is split exact.

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

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

is split exact, then is 0 -smooth over .

Next: Bibliography
2011-07-21