1
$\begingroup$

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

$\endgroup$

1 Answer 1

1
$\begingroup$

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 $X_2X_3\cdots X_n$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.