Hi,

I am reading this paper and have the following question:.

On page 5 you can see different types of $A_n$-modules ($\ A_n=k[x,y]/\langle x^2,y^{n+2},xy^{n+1}\rangle\ $)$\ $ and later in the paper it says that $DA_i^j$ is the dual module of $A_i^j$.

$A_i^j$ is an $A_n$-module with $k$-basis {$x,xy,...,xy^n,\ y^j,...,y^{n+1}$}, $A_n$ acts on $A_i^j$ by multiplication and we have $A_i^j\subseteq A_n$.

Question: Why does the shape of $DA_i^j$ look like what it looks like, that means, why do the pictures of $A_i^j$ and $DA_i^j$ on page 5 not coincide?

In the paper, $k$ need not be algebraically closed, but if it were, then I would think that you could compute $DA_i^j$ by the formula $DA_i^j=$Hom$_k(A_i^j,k)$.

If I do this, the shape of $DA_i^j$ looks like the shape of $A_i^j$, but that isn't the case in the paper.

Is Hom$_k(A_i^j,k)$ isomorphic to the module $DA_i^j$ in the paper, or, in general, in what different way do they compute the shape of $DA_i^j$?

Thanks for the help.