I am trying to classify all modules of length $4$ over the ring $A=\mathbb C[x,y]$, supported at the origin $0\in \mathbb C^2$, up to ($A$-linear) isomorphism. Let $\mathfrak m=(x,y)$ be the ideal of the origin and $k\simeq \mathbb C$ the residue field. Here is what I found so far:

- Structure sheaves of length $4$ subschemes $Z\subset \mathbb C^2$. These come into three types, $$B_1=A/(x,y^4),\qquad B_2=A/(x^2,y^2),\qquad B_3=A/(x^3,xy,y^2).$$
- $k^{\oplus 4}$,
- $k\oplus A/(x,y^3)$,
- $k\oplus A/\mathfrak m^2$,
- $k\oplus \textrm{Hom}_k(A/\mathfrak m^2,k)$,
- $k^2\oplus A/(x,y^2)$,
- $A/(x,y^2)\oplus A/(x,y^2)$,
- $\textrm{Hom}_k(B_3,k)$.

I think these are pairwise non-isomorphic.

Are these *all* the isomorphism types, or is any other module hiding? Also, I think this classification problem is not solved in general (for arbitrary length), does anyone know up to which length the classification is known?

Thanks!

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