Exercise 31, § 5

Let $k$ be a field. (a) Let $A$ be a noetherian, local, complete, regular $k$-algebra, whose residue field $\kappa_A$ is a separable extension of $k$. Now prove that $A$ is formally smooth and isomorphic to $\kappa_A[[T_1, ..., T_n]]$. (b) Let $B$ be a $k$-algebra of finite type, $p \in \mathrm{Spec}(B)$ a smooth point (p. 97, Exercise 23). Now prove that $\hat{B}_p$ is a formally smooth $k$-algebra. (c) Let $\kappa$ be a finite type extension of $k$ and let $n \in \mathbf{N}$. Now prove that there exists a formally smooth $k$-algebra with residue field $\kappa$ and dimension $n$. (d) Let $A$ be a formally smooth $k$-algebra whose residue field $\kappa_A$ is not a separable extension of $k$. Now prove that $A$ is not isomorphic as a $k$-algebra to $\kappa_A[[T_1, ..., T_n]]$.

**T 32) Let k be a field, K a finite type extension of k, n a positive integer, A and B two formally smooth $k$-algebras, of dimension n, such that the extensions $\kappa_A$ and $\kappa_B$ of k are isomorphic to K. We propose to show that A and B are isomorphic $k$-algebras. (a) Reduce to the case where K is a finite radicial extension of k (Exercise 29, a)). (b) Consider the completed algebra $C = A \otimes_k B$ of $A \otimes_k B$ with respect to the ideal

Now prove that C is a noetherian, local, complete and regular $k$-algebra. (For this last point, one will use the canonical homomorphism $\rho$ of A into C. We shall show that it makes C a flat A-module and that $C \otimes_A \kappa_A$ is isomorphic to $\kappa_A \otimes_k B$. One will then use Exercise 14, p. 96.) (c) Now prove that $\rho$ induces an isomorphism on the residue fields and an injection of $m_A / m_A^2$ into $m_C / m_C^2$. Construct then a retraction $\pi$ of $\rho$ which is a homomorphism of $k$-algebras. Now prove that $\pi \circ \rho' : B \to A$ is an isomorphism of $k$-algebras. (We have denoted by $\rho'$ the canonical homomorphism of B into C.)