Socle of Almost Complete Intersections

Question

Let $(A,m)$ be a complete Artinian local ring over a field $K$.
We focus on almost complete intersection ring $A$ of the form

$A = K[[X_1,...,X_N]]/(f_1,...,f_{N+1})$.

We assume that none of $f_i$ can be omitted and the Krull dimension of $A$ is zero.

We denote by $Soc(A)$ the socle of $A$, which is defined by $Soc(A)$ := Annihilators of $m$ in $A$. $Soc(A)$ is a vector space over $K$. We recall

Theorem (Kunz): $\dim_K Soc(A) > 1$, i.e., $A$ is NOT Gorenstein.

---

Question: Does the dimension of Soc(A) as the vector space over K go to

infinite as N -> infinite?

Please help me with this. Any information or reference is very much welcome.

Pierre MATSUMI

Answer 1

Your question is a little vague, but for my best guess at your meaning, the answer is: no, the socle dimension need not diverge to $\infty$. For each $N\geq 3$, define $f_1 = X_1^2$, $f_2 = X_1X_2$, and $f_{r+1} = X_r^2$ for $r=2,\dots,N$. Then the socle is the $2$-dimensional vector space spanned by the images of $X_1X_3\cdots X_N$ and $X_2X_3\cdots X_n$.