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