Example (Complete intersection formed by four coordinate planes of an affine space of dimension 4)

Let $k$ be a field. Let S denote the polynomial ring $k[T_1, T_2, T_3, T_4]$. Recall (VIII, § 2, No. 4, Theorem 3) that every maximal chain of prime ideals of S is of length 4. Let m denote the ideal of S generated by the $T_i$, a the ideal generated by $T_1T_2$ and $T_3T_4$, and $p_{ij}$, for $1 \leq i < j \leq 4$, the ideal generated by $T_i$ and $T_j$. The ideals $p_{ij}$ are prime of height 2, their sum is the maximal ideal m, and one has $a = p_{13} \cap p_{14} \cap p_{23} \cap p_{24}$.

a) The ring $A = S/a$ is reduced; the irreducible components of Spec(A) are the sets $X_{ij} = V(p_{ij}/a)$ for $i = 1,2,\ j = 3,4$, which are of dimension 2 and all contain the point $m/a$. In particular, Spec(A) is connected and of dimension 2. The intersection of two distinct components $X_{ij}$ and $X_{kl}$ is reduced to $\{m/a\}$ if $\{i,j\} \cap \{k,l\} = \varnothing$, of dimension 1 otherwise. It follows that the conclusion of cor. 4 is satisfied with $d = 1$ (we shall see later that A is a Macaulay ring, so that the assumption of corollary 4 is itself satisfied for $d = 1$).

b) Let b denote the ideal of S generated by $T_1T_2$, $T_1T_3$, $T_2T_4$, $T_3T_4$, and B the ring $S/b$. One has $b = p_{14}p_{23} = p_{14} \cap p_{23}$. The ring B is reduced. Its spectrum is identified with the closed subset $X_{14} \cup X_{23}$ of Spec(A); it has two irreducible components (of dimension 2) whose intersection is reduced to a point. The depth of B along this point is strictly positive because B is reduced, and less than 1 by Theorem 3, hence equal to 1.* Thus B is not a Macaulay ring. \*