Exercise 15, § 7

Let $A$ be a local, noetherian, complete ring, with maximal ideal m. Put $d = \dim(A)$. Suppose that A contains a subfield k such that $A = k + m$.

Prove that one can find $x_1, ..., x_d$ in m such that: $\alpha)$ the morphism $\varphi : k[[X_1, ..., X_d]] \to A$ defined by $\varphi(X_i) = x_i$ makes A an integral $k[[X_1, ..., X_d]]$-algebra ; $\beta)$ denoting by $m_0 = (X_1, ..., X_d)$ the maximal ideal of $k[[X_1, ..., X_d]]$, one has $\overline{m_0A} = m$, and hence $e_{m_0A}(A) = e_m(A)$. Prove that if $x_1, ..., x_d$ satisfy condition $\alpha)$, condition $\beta)$ is satisfied if and only if the initial forms $\xi_1, ..., \xi_d$ of the $x_i$ in $\mathrm{gr}_m(A)$ form a maximal transversal sequence.