1
$\begingroup$

Let $A \colon= {\Bbb F}_p[[X_1,\ldots,X_n]]$ be a $n$-variable power series ring over a finite field ${\Bbb F}_p$. We put ${\frak m}_A \colon= (X_1,\ldots,X_n)$.

Definition(Super-Gorenstein ideal): ${\frak a} = (F_1,...,F_d)$ is called a ‘‘super-Gorenstein ideal” of $A$, if for each $k \geq 1$ there exists an ideal ${\frak b}_k \in {\frak m}_A^k$ such that

  1. ${\frak b}_k$ has height $n$ and is generated as ${\frak b}_k = (f_{k,1},...,f_{k,n})$ with each $f_{k,i} \in {\frak m}_A^k$.

  2. ${\frak a} + {\frak b}_k = (F_1,...,F_d,f_{k,1},...,f_{k,n})$ has $F_1,...F_d,f_{k,1},...,f_{k,n}$ exactly as the generators of ${\frak a} + {\frak b}_k$. Namely if we denote by $\mu({\frak a} + {\frak b}_k)$ the number of generators of ${\frak a} + {\frak b}_k$, $\mu({\frak a} + {\frak b}_k)=d + n$.

  3. $A/({\frak a} + {\frak b}_k)$ is a Gorenstein local ring (of dimension $0$).

Question: Does a super-Gorenstein ideal ${\frak a}$ exist?

Improve this question
$\endgroup$
2
  • $\begingroup$ Isn't the $(0)$ ideal super-Gorenstein? What is a reference for this definition? And, what is the use of the characteristic of the coefficient ring regarding super-Gorensteinness? As you know almost complete intersections are not Gorenstein (Kunz). Therefore, at least when $\mathfrak{a}$ is a principal ideal, then it is not super-Gorenstein even though it is Gorenstein. $\endgroup$
    – Youngsu
    Commented Dec 27, 2013 at 2:46
  • $\begingroup$ Dear Youngsu, sorry. I should have used the terminology ‘‘Mock-Gorenstein Ideal” rather than Super Gorenstein Ideal. Do you guess there is any non-zero mock-Gorenstein Ideal? Pierre $\endgroup$
    – Pierre
    Commented Dec 27, 2013 at 20:10

0

Reset to default

You must log in to answer this question.