# When is $N^{*} \otimes_K M$ projective for a local Hopf algebra?

Given a finite dimensional local Hopf algebra $$A$$ over a field $$K$$ and two finite dimensional indecomposable modules $$N$$ and $$M$$. Is it known when the module $$N^{*} \otimes_K M$$ is projective? Can this happen when $$M$$ and $$N$$ are not projective? Can it happen for a fixed $$M$$ and $$N=\Omega^k(M)$$ for some $$k$$?

Special cases such as $$A$$ being $$KG$$ for a $$p$$-group $$G$$ are also welcome.

For group algebras, and certain other classes of Hopf algebras, there is a simple criterion in terms of (rank or cohomological) varieties of modules. The varieties $$V_G(M)$$ and $$V_G(N)$$ intersect trivially if and only if $$M^*\otimes N$$ is projective. This can easily happen without $$M$$ or $$N$$ being projective: for an example that doesn't need varieties, take $$G=C_p\times C_p$$ and $$K$$ of characteristic $$p$$, with $$M=K[G/(C_p\times1)]$$ and $$N=K[G/(1\times C_p)]$$.