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For questions on modules over rings.

10 votes
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If the tensor product of two $kG$-modules is projeсtive, does either of them have to be proj...

Then neither $M$ nor $N$ is projective (for modules over a $p$-group in characteristic $p$, projective is the same as free), but $M \otimes_k N \cong kG$. …
John Palmieri's user avatar