A theorem of Fong's type for graded algebras. (Q2574690)

Let \(\mathcal O\) be a complete discrete valuation ring with an algebraically closed residue field of characteristic \(p>0\), let \(G\) be a finite \(p\)-solvable group, and let \(A=\bigoplus_{x\in G}A_x\) be a \(G\)-graded \(\mathcal O\)-algebra which is free of finite rank as \(\mathcal O\)-module. The authors prove the following: (i) For any point (i.e. conjugacy class of primitive idempotents) \(\alpha\) of \(A\), there exists a Hall \(p'\)-subgroup \(H\) of \(G\) such that \(\alpha\cap A_H\neq\emptyset\) where \(A_H=\bigoplus_{x\in H}A_x\). (ii) Let \(H\) be a Hall \(p'\)-subgroup of \(G\), and let \(\beta\) be a point of \(A_H\). Then there is a point of \(A\) containing \(\beta\) if and only if for any Hall \(p'\)-subgroup \(H'\) of \(G\) every idempotent in \(\beta\cap A_{H'}\) is primitive in \(A_{H'}\).