Tensor product of finite global dimension algebras

Question

Given two finite dimensional connected algebras A and B over a field $K$ with finite global dimension. Their tensor product is not necessarily of finite global dimension when the field is not algebraically closed (seperable should be enough). Is there an example where the tensor product of $A$ and $B$ is neither selfinjective nor of finite global dimension?

Answer 1

Let $K=k(t^p)$, where $k$ has characteristic $p>0$.

Then $$k(t)\otimes _K\pmatrix{k(t)&k(t)\\0&k(t)}\cong \pmatrix{K[s]/(s^p)&K[s]/(s^p)\\0&K[s]/(s^p)}.$$