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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?

Given two finite dimensional 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?

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?

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Mare
Mare
  • 26.5k
  • 6
  • 25
  • 104

Tensor product of finite global dimension algebras

Given two finite dimensional 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?