Next: Bibliography

# Commutative algebra

Yoshifumi Tsuchimoto

DEFINITION 09.1   Let be a ring. an -module is said to be projective if it satisfies the following condition: For any -module morphism and for any surjective -module homomorphism , lifts'' to an -module morphism .

LEMMA 09.2   An -module is projective if and only if it is a direct summand of a free -module.

PROPOSITION 09.3   Let be a 0 -smooth algebra over a ring . Then is projective.

PROOF.. Let us express the algebra as a quotient where is a polynomial algebra and is an ideal of . Then by Theorem 08.5, we know that

is split exact. So is a direct sum of . On the other hand, is free -module so that is also a free -module.

We would like to define smoothness'' as a something good. Especially, we would expect smooth algebras'' to be flat. But that is not always true if we regard smoothness'' as 0 -smoothness. The following example is an easy case of [1, example 7.2].

EXAMPLE 09.4   Let us put and put

Then we see that . Thus is 0 -smooth over . where as is not flat over .

DEFINITION 09.5   Let be a ring.
1. An -algebra is said to be finitely generated over if is generated by a finite set as an -algebra. In other words, it is finitely generated if there exists a surjective -algebra homomorphism from a finitely generated polynomial ring to .
2. An -algebra is said to be finitely presented over if there exists a surjective -algebra homomorphism from a finitely generated polynomial ring to such that its kernel is a finitely generated ideal of . is a finitely generated ideal of .

DEFINITION 09.6   Let be a ring. An -algebra is said to be smooth over if it is 0 -smooth and finitely presented over .

We may define unramified/étale algebras in a same manner.

Let us recall the definition of Noetherian ring.

DEFINITION 09.7   A ring is called Noetherian if its ideals are always finitely generated.

PROPOSITION 09.8

If is Noetherian, then:

1. Any of its quotient ring is Noetherian.
2. The polynomial ring is Noetherian.
It follows that any finitely generated -algebra is also Noetherian. We note also that is finitely presented over in this case.

Next: Bibliography
2011-07-21