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Heskie
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If the tensor product of two $kG$-modules is projetive, does either of them have to be projective?

Let $G$ be a finite group and $k$ a field, let us assume that char($k$) divides the group order. Let $kG$-mod denote the category of fintely generated $kG$-modules. This category has as a tensor product $\otimes_{k}$ with diagonal $G$-action. Given now $M,N\in kG$-mod such that $M\otimes_{k}N$ is projective, can we then conclude that either $M$ or $N$ had to be projective? If not, can we ask for certain conditions on the field $k$ or the group $G$ such that the statement holds?

Heskie
  • 113
  • 1
  • 4