Let $A$ be a local selfinjective algebra with indecomposable module $M$. Let $N=A \oplus M$.

When there is an indecomposable module $U$ not in $add(N)$, having finite $add(N)$-resolution for some choice of $A$ and $M$?

This is not possible in general due to the following examples:

• $A=K[x]/(x^n)$ for arbitary $M$

• $A$ arbitrary and $M$ simple.