Blocks of normal subgroups and Morita equivalences (Q2706768)

Let \(G\) be a finite group, let \((K,{\mathcal O},F)\) be a suitable \(p\)-modular system, and let \(R\) be a \(G\)-graded \(\mathcal O\)-algebra which is a crossed product. Every block \(b\) of the identity component \(R_1\) of \(R\) determines a Clifford extension \(E(b)\), a twisted group algebra of a subgroup \(G[b]\) of \(G\) over \(F\). Similarly, every simple \(K\otimes_{\mathcal O}R_1\)-module \(X\) determines a Clifford extension \(E(X)\), a twisted group algebra of a subgroup \(G_X\) of \(G\) over \(K\), and every indecomposable \(F\otimes_{\mathcal O}R_1\)-module \(U\) determines a Clifford extension \(E(U)\), a twisted group algebra of a subgroup \(G_U\) of \(G\) over \(F\). In the first part of the paper, the author shows that these Clifford extensions are invariants of the graded Morita equivalence class of \(R\). In the second part of the paper, the author gives a graded version of a result of \textit{M. Linckelmann} [Turk. J. Math. 22, No. 1, 93-107 (1998; Zbl 0922.20017)] concerning splendid Morita equivalences between blocks with isomorphic defect groups and equivalent Brauer categories.