The generating hypothesis for the stable module category of a \(p\)-group.
Let \(k\) be a field of characteristic \(p>0\), and let \(G\) be a nontrivial finite \(p\)-group. For finite-dimensional \(kG\)-modules \(M,N\), a map \(\varphi\in\Hom_{kG}(M,N)\) is called a ghost if the induced map in Tate cohomology \(\widehat H^i(G,M)\to\widehat H^i(G,N)\) is trivial for each \(i\). A ghost \(\varphi\) is called trivial if \(\varphi\) factors through a projective \(kG\)-module. The authors prove that all ghosts for \(G\) are trivial if and only if \(G\) is cyclic of order 2 or 3. This result is related to Freyd's generating hypothesis in stable homotopy theory.
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