Remark 2

Let $x_1, ..., x_n$ be elements of A, and put $H_i = V(x_i)$ for $1 \leq i \leq n$. Suppose there exists a finite-type A-module M with support $V = \mathrm{Spec}(A)$, such that $(x_1, ..., x_n)$ is a completely secant sequence for M. Then every irreducible component of $H_1 \cap ... \cap H_n$ is of codimension n in V: this follows easily from the corollary to Prop. 3 of No. 2.