Proposition 2

Let $A$ be a noetherian local ring, $x = (x_1, ..., x_r)$ a sequence of elements of $m_A$ and $x$ the ideal generated by $x$. The following conditions are equivalent:

(i) the ring $A$ is regular, and $x$ forms part of a system of coordinates of $A$; (ii) the ring $A/x$ is regular and $x$ is a secant sequence for $A$; (iii) the ring $A/x$ is regular and $x$ is a completely secant sequence for $A$.

Moreover, when these conditions are satisfied, $x$ is a prime ideal of $A$.

(iii) $\Rightarrow$ (ii): this follows from the corollary of Prop. 3 of § 3, No. 2.

(ii) $\Rightarrow$ (i): suppose that $x$ is a secant sequence for $A$ and that the noetherian local ring $A/x$ is regular. Let $(x_{r+1}, ..., x_d)$ be a sequence of elements of $A$ whose classes modulo $x$ form a system of coordinates of $A/x$. Then the sequence $(x_1, ..., x_d)$ generates the ideal $m_A$ of $A$, and we have

It follows that $A$ is regular and that $(x_1, ..., x_d)$ is a system of coordinates of $A$.

(i) $\Rightarrow$ (iii): if condition (i) is satisfied, the sequence $x$ is completely secant (No. 2, Th. 1), hence secant by the corollary of Prop. 3 of § 3, No. 2. We therefore have

(3)

moreover, the classes of $x_1, ..., x_r$ modulo $m_A^2$ are linearly independent over the field $\kappa_A$, and therefore we have

(4)

Formulas (3) and (4) show that $A/x$ is regular.

Every regular noetherian local ring is an integral domain by Cor. 1 of Th. 1 of No. 2. It follows that $x$ is prime if $A/x$ is regular.