Exercise 7, § 6

a) Now prove that an integral Dedekind ring of characteristic 0 is a Grothendieck ring.

b) Let $k$ be a field of characteristic $p > 0$ such that $k$ is of infinite dimension over $k^p$, $r$ an integer $\geqslant 1$, and $A$ the subring of $k[[X_1, \ldots, X_r]]$ formed by the formal series whose coefficients generate a finite-dimensional vector space over $k^p$. Prove that $A$ is a regular local ring of dimension $r$ whose completion is identified with $k[[X_1, \ldots, X_r]]$, but is not a Grothendieck ring (use Exercise 17 of III, § 3).