Theorem 3

Let A be a local, noetherian and complete ring whose residue field k is of characteristic p. Let C be a p-ring of infinite length, whose residue field is isomorphic to k (No. 3, Prop. 5).

a) Let m be the dimension of the vector space $m_A/(m_A^2 + pA)$ over the field k. There exists an ideal $a$ of the ring $C[[T_1, ..., T_m]]$ such that A is isomorphic to $C[[T_1, ..., T_m]]/a$.

b) Let d be the dimension of A. Suppose that $p1_A$ is not a divisor of 0 in A. Then there exists a subring $A'$ of A isomorphic to $C[[T_1, ..., T_{d-1}]]$ and such that A is a finite algebra over $A'$.

Let $C'$ be a Cohen subring of $A$ (No. 2, Theorem 1). Since $C$ is of infinite length, there exists a homomorphism of $C$ onto $C'$ (No. 3, Proposition 4). Consequently, there exists a local homomorphism $u : C \to A$. Let us choose elements $t_1, ..., t_m$ of $m_A$ whose classes form a basis of the vector space $m_A/(m_A^2 + pA)$ over the field $k$. We have $u(p1_C) = p1_A$, and Lemma 3, b) proves the existence of a surjective homomorphism of $C[[T_1, ..., T_m]]$ onto $A$, extending $u$ and mapping $T_i$ to $t_i$ for $1 \leq i \leq m$. This proves a).

Suppose that $p1_A$ is not a divisor of $0$ in $A$, hence is secant for $A$ (VIII, § 3, No. 2, Proposition 3). Then there exist (VIII, § 3, No. 2, Theorem 1) elements $t_1, ..., t_{d-1}$ of $m_A$ such that the sequence $(p1_A, t_1, ..., t_{d-1})$ is a maximal secant sequence for $A$. The noetherian local ring $C$ is regular, and $(p1_C)$ is a system of coordinates of $C$. The assertion b) of Theorem 3 then follows from Lemma 4, b).