I have this problem about a property of Gorenstein artinian rings:
Let $(A,m)$ be an artinian local Gorenstein graded ring such that $A_s\neq 0$ and $A_{s+1}=0$ where $A_i$ is the degree $i$ part of $A$. Let $I$ be an homogeneus ideal of $A$ and $J=(0:I)$. Show that $$ (0:I_t)_{s-t} = J_{s-t} $$
A containment is obvius: since $I_t\subset I$ then $(0:I_t)_{s-t} \supset J_{s-t}$.
But i can't figure out the viceversa (maybe it is false?). Here some remarks:
- $A_{s+k}=0$ for all $k>1$
- $\text{dim}_k (A_t) = \text{dim}_k(A_{s-t})$ with $k=A/m$ (hence $A_s=k$)
- The multiplication map $A_t \times A_{s-t}\rightarrow A_s=k$ is a perfect pairing for all $t$
Thank you in advance!