Exercise 6, § 5

It is said that a noetherian ring $A$ is regular if, for every prime ideal $p$ of $A$, the local ring $A_p$ is regular. a) Let $A$ be a noetherian ring. Then $A$ is regular if and only if $A_m$ is regular for every maximal ideal $m$ of $A$ (use Exercise 5). b) Let $A$ be a noetherian ring. Prove that $A$ is finite-dimensional regular if and only if $\mathrm{dh}(A)$ is finite (use Exercise 4). c) Prove that the ring described in Exercise 13, p. 83 is regular of infinite dimension. d) Let $A$ be a noetherian regular ring, $S$ a multiplicative subset. Prove that $S^{-1}A$ is regular. e) Let $A$ be a noetherian regular ring. Prove that $A[X]$ is regular. (Reduce by localizing to the case where $\mathrm{dh}(A)$ is finite. Use A, X, p. 143, Theorem 1, and b).)

$^1$ For complements concerning this exercise, one may consult R. Stanley, The Hilbert function of a graded algebra, Advances in Math., 28 (1978), p. 57-83.

EXERCISES

AC VIII.95