A remark on the Puig correspondence and Thévenaz's lifting theorem (Q1579098)

Let \(p\) be a prime, let \(R\) be a suitable \(p\)-adic ring, and let \(H\) be a subgroup of a finite group \(G\). The author relates the question whether an indecomposable \(RG\)-module \(V\) is induced from an indecomposable \(RH\)-module \(W\) to the corresponding question for their Green correspondents \(V'\) and \(W'\), respectively. The main result is formulated in a more general setting. Thus let \(A\) be a \(G\)-algebra over \(R\), and suppose that there are inclusions of pointed groups \(P_\gamma\leq N_\nu\leq G_\alpha\) and \(P_\gamma\leq M_\mu\leq H_\beta\) on \(A\) where \(P_\gamma\) is a defect pointed subgroup of \(G_\alpha\), \(N=N_G(P_\gamma)\) and \(M=N_H(P_\gamma)\). It is shown that \(N_\nu\) is relatively \(M_\mu\)-free if \(G_\alpha\) is relatively \(H_\beta\)-free. The converse holds if \(G\) contains a normal subgroup \(U\supseteq P\) such that \(\gamma_\infty(U)\subseteq H\). Here \(G_\alpha\) is called relatively \(H_\beta\)-free if there are idempotents \(e\in\alpha\), \(f\in\beta\) such that \(f({^gf})=0\) for \(g\in G\setminus H\) and \(e=\text{Tr}^G_H(f)\).