Stable Clifford extensions of modules. (Q854909)

This short paper proves a simple result stated by E. Cline almost 40 years ago. \(G\) is a finite group, and \(R\) is a ring of characteristic 0. \((K,O,F)\) is a \(p\)-modular system, where \(O\) is a complete discrete valuation ring, \(K\) is the quotient field of \(R\), \(F\) is the residue field of \(O\) with characteristic \(p>0\). In particular, it is assumed that \(R\) is either \(F\) or \(O\) for big enough \(F\) and \(K\). The result is, if \(H\) is a normal subgroup of \(G\), and \(W\) is a \(G\)-invariant indecomposable \(RH\)-module with vertex \(Q\) and \(W'\) is the Green correspondent of \(W\) with respect to \((H,Q,N_H(Q))\), then \(G=HN_G(Q)\). Moreover, under the canonical isomorphism \(G/H\cong N_G(Q)/N_H(Q)\), \(E/J(\Lambda)\cong E'/J(\Lambda')E'\) as twisted group algebras, where \(E'=\text{End}_{RN_G(Q)}((W')^{N_G(Q)})\) and \(\Lambda'=\text{End}_{RN_H(Q)}(W')\). A corollary is that \(W\) can extend to an \(H\)-projective \(RG\)-module if and only if \(W'\) can extend to an \(N_H(Q)\)-projective \(RN_G(Q)\)-module.