**Yoshifumi Tsuchimoto**

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].

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

- 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 . - 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 .

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

Let us recall the definition of Noetherian ring.

*If
is Noetherian, then:
*

- Any of its quotient ring is Noetherian.
- The polynomial ring is Noetherian.