Theorem 2

Let $A$ be a complete regular local noetherian ring whose residue field is of characteristic $\neq 0$, and let $V$ be a Cohen subring of $A$. Put $n = \dim(A)$.

a) Suppose that $V$ is a field. Then the $V$-algebra $A$ is isomorphic to the algebra of formal series $V[[X_1, ..., X_n]]$.

b) Suppose that $V$ is a complete discrete valuation ring and that one has $m_V \not\subset m_A^2$. Then the $V$-algebra $A$ is isomorphic to the algebra of formal series $V[[X_1, ..., X_{n-1}]]$.

c) Suppose that $V$ is a complete discrete valuation ring and that one has $m_V \subset m_A^2$. Then there exists a special polynomial $P$ in $V[[X_1, ..., X_{n-1}]] [T]$ and a $V$-isomorphism of $A$ onto $V[[X_1, ..., X_{n-1}]] [T]/(P)$.

Assertion a) follows at once from Cor. 3 of Th. 1 of No. 2.

Let us prove b). Let $(x_1, ..., x_m)$ be a sequence of elements of $m_A$, and let $\varphi_0$ be the homomorphism of $V[X_1, ..., X_m]$ into $A$ which coincides with the identity on $V$ and sends $X_i$ to $x_i$ for $1 \leq i \leq m$. If $a$ is the ideal of $V[X_1, ..., X_m]$ generated by $X_1, ..., X_m$, one has $\varphi_0(a) \subset m_A$, hence $\varphi_0$ extends by continuity to a homomorphism $\varphi$ of $V_m = V[[X_1, ..., X_m]]$ into $A$. Let $\pi$ be a uniformizer of $V$. By Corollary 2 of Theorem 1 of No. 2, $\varphi$ is an isomorphism of $V_m$ onto $A$ if and only if $(\pi, x_1, ..., x_m)$ is a system of coordinates of $A$. But the canonical mapping of $m_V/m_V^2$ into $m_A/m_A^2$ is injective, since one has $m_V \not\subset m_A^2$ and $m_V/m_V^2$ is of rank 1 over $\kappa_V$. Hence $m_A/(m_V + m_A^2)$ is of rank $n - 1$ and $(\pi, x_1, ..., x_m)$ is a system of coordinates of $A$ if and only if the classes of the $x_i$ form a basis of $m_A/(m_V + m_A^2)$ over $\kappa_V$. Hence b).

Let us prove c). Let $(y_1, ..., y_n)$ be a system of coordinates of $A$. As above, put $V_n = V[[Y_1, ..., Y_n]]$ and consider the homomorphism $\varphi$ of $V_n$ into $A$ which coincides with the identity on $V$ and sends $Y_i$ to $y_i$ for $1 \leq i \leq n$. Then $\mathrm{gr}(\varphi)$ is surjective, hence $\varphi$ is surjective (III, § 2, no 8, Corollary 2 of Theorem 1). The kernel $p$ of $\varphi$ is a prime ideal of $V_n$; since one has

the prime ideal $p$ is of height 1 (§ 1, no 3, Proposition 8). But the ring $V_n$ is factorial by Proposition 8 of VII, § 3, no 9; consequently, the ideal $p$ is principal (VII, § 3, no 2, Theorem 1).

Let $R$ be a generator of the ideal $p$ of $V_n$. By Lemma 3 of VII, § 3, No. 7, there exist integers $u(1), ..., u(n-1)$ at least equal to 1, and an isomorphism $\sigma$ of $V[[X_1, ..., X_{n-1}, T]]$ onto $V_n$ such that

and such that $\sigma^{-1}(R) = Q$ satisfies $Q(0, ..., 0, T) \neq 0$. Moreover, by the preparation theorem (VII, § 3, No. 8, Prop. 6), there exists a polynomial P in V[[X_1, ..., X_{n-1}]] [T] of the form

generating the same ideal as Q in V[[X_1, ..., X_{n-1}, T]], and such that $a_i(0, ..., 0) \in m_V$ for $1 \leq i \leq d$. It is deduced that A is V-isomorphic to V[[X_1, ..., X_{n-1}, T]]/(P). But V[[X_1, ..., X_{n-1}, T]] is the direct sum of the ideal (P) and of the V[[X_1, ..., X_{n-1}]]-module with basis 1, T, ..., $T^{d-1}$ (VII, § 3, No. 8, Prop. 5); hence A is V-isomorphic to V[[X_1, ..., X_{n-1}]] [T]/(P).

Put C = V[[X_1, ..., X_{n-1}]] and denote by α the class of $a_d(X_1, ..., X_{n-1})$ modulo $m_C^2$. We have $\kappa_V = \kappa_C = \kappa_A$. By Prop. 4 of No. 4, the kernel of the canonical homomorphism from $m_C/m_C^2$ into $m_A/m_A^2$ is equal to $\kappa_C \alpha$. Since the image of $m_V/m_V^2$ in $m_A/m_A^2$ is zero and $m_V/m_V^2$ is of rank 1 over $\kappa_C$, it follows that $a_d$ belongs to $m_V + m_C^2$, but not to $m_C^2$. Consequently, the polynomial P is special, as was to be proved.

\* Remark. — Let k be a field, A a complete regular local noetherian k-algebra. When $\kappa_A$ is not a separable extension of k, it is not true in general that A is isomorphic as a k-algebra to $\kappa_A[[T_1, ..., T_n]]$ where $n = \dim(A)$ (p. 98, Exercise 29). \*