On defect groups of interior \(G\)-algebras and vertices of modules
Let \(G\) be a finite group, and let \({\mathcal O}\) be a complete discrete valuation ring with algebraically closed residue field of characteristic \(p\neq 0\). Recall that an interior \(G\)-algebra over \({\mathcal O}\) is a pair consisting of an \({\mathcal O}\)-order \(A\) and a unitary homomorphism \(\rho: {\mathcal O}G\to A\). Then every \(A\)-module can be considered as an \({\mathcal O}G\)-module via \(\rho\). Conversely, the author says that an \({\mathcal O}G\)- module \(V\) belongs to \(A\) if \(V\) is isomorphic to a direct summand of \(\hbox{Res}_{{\mathcal O}G}^ A(W)\) for some \(A\)-module \(W\). (This generalizes the usual situation where \(A\) is a block of \({\mathcal O}G\).) Suppose now that \(A^ G:=\{a\in A: \rho(g)a=a\rho(g)\hbox{ for }g\in G\}\) is a local \({\mathcal O}\)-order. Then, by Green's theory, a defect group \(D\) of \(A\) is defined; \(D\) is a \(p\)-subgroup of \(G\). The author shows that, under suitable additional hypotheses, there exists an indecomposable \({\mathcal O}G\)-module with vertex \(D\) belonging to \(A\).
scientific article; zbMATH DE number 7622