Example 5

Let us keep the preceding notation. A polynomial P in $V_n[T]$ of degree $d \geq 2$ is said to be special if it is of the form $T^d + \sum_{i=1}^d a_i T^{d-i}$, where $a_1, ..., a_{d-1}$ belong to $m_{V_n}$, $a_d$ belongs to $m_V + m_{V_n}^2$ but not to $m_{V_n}^2$. In particular, P is an Eisenstein polynomial for $m_{V_n}$. Put $A = V_n[T]/(P)$. By the corollary of Prop. 4 of No. 4, the ring A is complete regular local noetherian, of dimension $n + 1$. If t is the class of T modulo (P), the sequence $(1, t, ..., t^{d-1})$ is a basis of the $V_n$-module A, and $(X_1, ..., X_n, t)$ is a system of coordinates of A: in fact, there exists a uniformiser $\pi$ of V such that $a_d \equiv \pi \mod. m_{V_n}^2$; since one also has

one has $\pi \in m_A^2$; since $m_A$ is generated by $\{\pi, X_1, ..., X_n, t\}$, this proves our assertion. Moreover, V is a Cohen subring of A, for $\kappa_V$ is identified with $\kappa_{V_n}$, and $\kappa_{V_n}$ with $\kappa_A$ by the corollary to Prop. 4 of No. 4.