Exercise 21, § 5

Let $I$ be an ordered set, $(A_i, \varphi_{ij})$ an inductive system of rings with inductive limit $A$. Suppose that, for all $i$, the ring $A_i$ is noetherian, that for every pair $(i, j)$ such that $i < j$, $\varphi_{ij}$ makes $A_j$ a flat $A_i$-module and that $A$ is noetherian. Now prove that if the $A_i$ are regular for all $i \in I$, then $A$ is regular. (One may first reduce to the case where the $A_i$ and $A$ are local and the $\varphi_{ij}$ are local. One will then use Exerc. 3, p. 94 and 14, p. 96.)