Exercise 10, § 7

Let $A$ be a complete noetherian local ring, $K$ its field of fractions. Let $\mathfrak{p}$ be a prime ideal of $A$; denote by $B$ the local ring $A_\mathfrak{p}$ and by $\widehat{B}$ its completion. a) Suppose that $A$ is regular and that $K$ has characteristic zero. Now prove that the $K$-algebra $K \otimes_B \widehat{B}$ is absolutely regular. b) Suppose that $A$ is regular and that $K$ has characteristic $p > 0$, so that $A$ is identified with a formal power series ring $k[[T_1, \ldots, T_n]]$ (IX, § 3, No. 3, Theorem 2). Denote by $(k_\lambda)_{\lambda \in A}$ the family of subfields $k_\lambda$ of $k$ containing $k^p$ and such that $k$ is a finite degree extension of $k_\lambda$. Let $\lambda \in A$; denote by $A_\lambda$ the ring $k_\lambda[[T_1^p, \ldots, T_n^p]]$, by $B_\lambda$ the local ring of $A_\lambda$ at the prime ideal $\mathfrak{p} \cap A_\lambda$, and by $K_\lambda$ the field of fractions of $A_\lambda$ and $B_\lambda$. Prove that the $A$-algebra $B$ (endowed with the discrete topology) is formally smooth relative to $B_\lambda$ (one may first prove that $\widehat{B}$ is isomorphic to $B \otimes_{B_\lambda} \widehat{B_\lambda}$). c) Under the hypotheses of $b$), prove that every local ring of $K \otimes_B \widehat{B}$ is of the form $(\widehat{B})_r$, with $r \in \mathrm{Spec}(\widehat{B})$ and $r \cap B = \{0\}$, and that it is formally smooth over $K$ relative to $K_\lambda$. Deduce that the $K$-algebra $K \otimes_B \widehat{B}$ is absolutely regular (prove that the family $\{K_\lambda\}_{\lambda \in A}$ of subfields of $K$ is filtrante decreasing and that $\bigcap_{\lambda \in A} K_\lambda = k^p((T_1, \ldots, T_n))$); then apply exercise 9). d) Return to the general case. Prove that $A$ is a Grothendieck ring (§ 6, exercise 5); if $q \in \mathrm{Spec}(B)$, it is a matter of proving that the $\kappa(q)$-algebra $\kappa(q) \otimes_B \widehat{B}$ is absolutely regular; replacing $A$ by $A/(A \cap q)$, one may suppose that $A$ is integral and $q = \{0\}$; one will use IX, § 2, No. 3, Theorem 3 and § 3, No. 3, Theorem 2 to reduce to the case where $A$ is a complete regular local ring.