Lemma 4

Suppose that the noetherian local ring C is regular, and that $(p_1, ..., p_m)$ is a system of coordinates of C (VIII, § 5, No. 1, Def. 1).

a) If the sequence $(u(p_1), ..., u(p_m), t_1, ..., t_n)$ is secant for A (VIII, § 3, No. 2, Def. 1), the homomorphism $v : B \to A$ is injective.

b) In order that v be injective and make A a finite algebra over B, it is necessary and sufficient that $(u(p_1), ..., u(p_m), t_1, ..., t_n)$ be a maximal secant sequence for A. Then A is of dimension $m + n$.

In order that the sequence $(u(p_1), ..., u(p_m), t_1, ..., t_n)$ be a maximal secant sequence for A, it is necessary and sufficient that it generate an ideal of definition of A, and that A be of dimension $m + n$ (VIII, § 3, No. 2, Th. 1). By lemma 3, c), this comes to saying that A is a finite B-algebra, and a ring of dimension $m + n$. Now C is a noetherian integral ring of dimension $m$, hence $B = C[[T_1, ..., T_n]]$ is a noetherian integral ring of dimension $m + n$ (VIII, § 3, No. 4, cor. 3 of Prop. 8). If A is a finite B-algebra, and if $a$ is the kernel of $v$, one has $\dim(A) = \dim(B/a)$ (VIII, § 2, No. 3, th. 1, c)); since B is an integral ring of finite dimension, one has $\dim(B/a) < \dim(B)$ if $a \neq \{0\}$ (VIII, § 1, No. 3, prop. 6, e)). Hence, if A is a finite B-algebra, $v$ is injective if and only if A is of dimension $m + n$. This proves b).

Suppose that the sequence $(u(p_1), ..., u(p_m), t_1, ..., t_n)$ of elements of $m_A$ is secant. One can adjoin to it (VIII, § 3, No. 2, Th. 1) elements $t_{n+1}, ..., t_{n+r}$ of $m_A$ in order to make it into a maximal secant sequence. From what precedes, there then exists an injective homomorphism $w$ of $C[[T_1, ..., T_n, T_{n+1}, ..., T_{n+r}]] = B[[T_{n+1}, ..., T_{n+r}]]$ which extends $v$ and maps $T_{n+j}$ to $t_{n+j}$ for $1 \leq j \leq r$. Hence $v$ is injective. This proves a).