Exercise 5, § 6

It is said that A is a Grothendieck ring if A is a noetherian ring and if, for every prime ideal $\mathfrak{p}$ of A, the completion of the local ring $A_\mathfrak{p}$ is an absolutely regular $A_\mathfrak{p}$-algebra (exer. 3).

a) Prove that every ring of fractions and every quotient of a Grothendieck ring is a Grothendieck ring.

b) Let A be a noetherian ring and B a faithfully flat absolutely regular A-algebra. If B is a Grothendieck ring, prove that the same is true of A (if $\mathfrak{p}$ is a prime ideal of A and q a prime ideal of B over $\mathfrak{p}$, one will show that $\widehat{B_q}$ is a faithfully flat $\widehat{A_\mathfrak{p}}$-module by using III, § 5, No. 4, Prop. 4; one will then apply exer. 3).