Exercise 13, § 7

Let $k$ be a field and $A$ a $k$-algebra essentially of finite type. Now prove that $A$ is a Grothendieck ring (we shall reduce to the case where $A$ is a local ring of a polynomial ring over $k$ in a finite number of indeterminates; using the preceding exercise, we shall then reduce to proving that, for every prime ideal $r$ of $A[T_1, ..., T_n]$, every local ring $C$ of $A[T_1, ..., T_n]$ at a maximal ideal containing $r$, and every ideal $q$ of the completed ring $\widehat{C}$ such that C ∩ q = rC, the ring $(\widehat{C})_q / r(\widehat{C})_q$ is regular. For this one may use Theorem 3 of No. 9).