Exercise 5, § 3

Let A be a noetherian local ring containing a field k, and let m be the maximal ideal of A. We suppose that A is a formally smooth k-algebra (the field k being discrete). (a) Let K be a field of representatives of the ring $A/m^2$ (exerc. 4, b)), and let $x_1, ..., x_d$ be elements of m whose images form a basis of the K-vector space $m/m^2$. Let K

be the ring of polynomials in d variables $X_1, ..., X_d$, n its ideal generated by $X_1, ..., X_d$, and φ the homomorphism from K into $A/m^2$ which associates $x_i$ to $X_i$ for $1 \leq i \leq d$ and induces the identity on K. Then φ induces an isomorphism $\overline{\varphi}$ of K/$n^2$ onto $A/m^2$. (b) For every integer $n \geq 1$, there exists a homomorphism $\psi_n : A \to K[X_1, ..., X_d]/n^{n+1}$ which, by passing to the quotients, induces $\overline{\varphi}^{-1}$. Such a homomorphism $\psi_n$ is surjective. (c) For every integer $n \geq 1$, the length of the A-module $A/m^{n+1}$ is at least $\binom{d+n}{d}$ and one has $\dim(A) \geq d$. (d) For every extension $k'$ of finite degree of k, and every maximal ideal $m'$ of the semi-local ring $A \otimes_k k'$, the local ring $(A \otimes_k k')_{m'}$ is regular.