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 $N=K[G/(1\times C_p)]$.
