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 .
We may define unramified/étale algebras in a same manner.
Let us recall the definition of Noetherian ring.
If is Noetherian, then: