Exercise 3, § 6

Let A be a noetherian ring and $\rho : A \to B$ a homomorphism of rings. We say that the A-algebra B is absolutely regular if B is a noetherian ring and a flat A-module and if, for every $\mathfrak{p} \in \mathrm{Spec}(A)$, the $\kappa(\mathfrak{p})$-algebra $\kappa(\mathfrak{p}) \otimes_A B$ is absolutely regular.

a) Let T be a multiplicative subset of B and S a subset of A such that $\rho(S) \subset T$. If B is a regular A-algebra, $T^{-1}B$ is a regular $S^{-1}A$-algebra.

b) For a noetherian A-algebra B to be absolutely regular, it is necessary and sufficient that, for every maximal ideal n of B, the $A_{\rho^{-1}(n)}$-algebra $B_n$ be absolutely regular.

c) If C is an absolutely regular B-algebra and B an absolutely regular A-algebra, C is an absolutely regular A-algebra.

d) If B is a noetherian ring and C is a faithfully flat B-module and an absolutely regular A-algebra, B is an absolutely regular A-algebra.

e) Let B be an absolutely regular A-algebra and $A'$ an A-algebra essentially of finite type. The $A'$-algebra $A' \otimes_A B$ is absolutely regular.