Hi,


I have two commutative finite-dimensional $k$-Algebras $A$ and $B$ ($k$ is a field).


I wonder, whether there is a way to get the finite-dimensional indecomposable modules of $A \otimes\_k B$,  

if I know the finite-dimensional indecomposable modules of $A$ and $B$.


For example, is there a way to do this, if $A=k[x]/(x^n)$ and $B=k[y]/(y^n)$?


Thank you very much.



Indecomposable modules of a tensor product of two algebras
No, in general there isn't. In your particular case by accident for $n=1$ this is possible. But even for $n=2$, there are only $2$ indecomposable $A$-modules up to isomorphism. But Kronecker already clasified the $A\otimes\_k B$-modules and there are infinitely many.
For $n>2$ there are still only finitely many indecomposable $A$-modules (namely $n$) up to isomorphism, but the situation is in some sense worse. The indecomposable $A\otimes\_k B$-modules are not even classifiable (one says $A\otimes B$ is wild).


