A non-example of a graded Frobenius algebra

Question

Take the class of finite dimensional graded algebras $A = \sum_i A_i$ satisfying $|A_n| = 1$ where $A_n \neq 0$ and $A_m = 0$, for all $m > n$. What is an example in this class that is not Frobenius? In another words an example such that the obvious bilinear pairing is degenerate.

Answer 1

A pretty trivial example is an algebra with the zero product. Less trivial is $k[x,y]/(x,y)^2$ where $x$ has degree $1$ and $y$ has degree $2$. One can make up various similar examples, e.g., $k[x,y,z]/(x^2,xy,xz,yz,y^3,z^2)$ where $\deg x = 1$, $\deg y = 2$, $\deg z = 3$. An example that's generated in degree $1$ is

$$ A = k[x,y]/(x^2,xy^2,y^4) = \operatorname{span}\{1,x,y,xy,y^2,y^3\} $$

Here $x \in A_1$ is nonzero, but $xA_2 = 0$. Incidentally, this example also has a symmetric Hilbert function ($\dim A_k = \dim A_{n-k}$ for all $k$), so having that isn't sufficient to be a Frobenius algebra, either.