Exercise 12, § 7

Let $A$ be a noetherian local ring. In order that $A$ be a Grothendieck ring, it is necessary and sufficient that, for every finite integral $A$-algebra $B$, every maximal ideal $m$ of $B$ and every prime ideal $q$ of the completed ring $\widehat{B_m}$ satisfying $B_m \cap q = (0)$, the local ring $(\widehat{B_m})_q$ be regular (to show that this condition is necessary, it suffices, thanks to the preceding exercise, to show that, for every prime ideal $\mathfrak{p}$ of $A$, and every finite extension $K$ of $\kappa(\mathfrak{p})$, the ring $K \otimes_A \widehat{A}$ is regular; one may introduce a finite integral $A$-algebra $B$ with field of fractions K, denote by m_1, ..., m_r its maximal ideals, and interpret the local rings of K ⊗_A Ā as local rings of one of the $\widehat{B_{m_i}}$.