Relative projectivity and relative endotrivial modules.
Let \(G\) be a finite group, \(k\) an algebraically closed field of characteristic \(p>0\) and \(V\) a finitely generated \(kG\)-module. A finitely generated \(kG\)-module \(M\) is called relatively \(V\)-projective if \(M\) is a direct summand of \(V\otimes N\) for some finitely generated \(kG\)-module \(N\). This notion generalizes the relative projectivity with respect to subgroups. Now suppose that \(V\) is absolutely \(p\)-divisible, i.e. \(p\) divides \(\dim U\) for every direct summand \(U\) of \(V\). (Otherwise every finitely generated \(kG\)-module is relatively \(V\)-projective.) The author calls a finitely generated \(kG\)-module \(M\) relatively \(V\)-endotrivial if \(\Hom_k(M,M)\cong k\oplus P\) where \(P\) is a relatively \(V\)-projective \(kG\)-module. This notion generalizes the classical notion of an endotrivial \(kG\)-module. The author investigates the behavior of relatively endotrivial modules under restriction, induction, inflation, the Green correspondence and the Heller functor. Every relatively \(V\)-endotrivial \(kG\)-module \(M\) has a unique indecomposable direct summand \(M_0\) which is also relatively \(V\)-endotrivial. Two relatively \(V\)-endotrivial \(kG\)-modules \(M,N\) are called equivalent if \(M_0\) and \(N_0\) are isomorphic. The tensor product induces the structure of an Abelian group on the set \(T_V(G)\) of equivalence classes of relatively \(V\)-endotrivial \(kG\)-modules. The author studies the behavior of \(T_V(G)\) under restriction and inflation. In the case where \(G\) is a finite \(p\)-group, the Dade group \(D(G)\) of capped endo-permutation \(kG\)-modules can be embedded into \(T_{\widetilde V}(G)\), for \(\widetilde V:=\bigoplus_{H<G}\text{Ind}_H^G(k)\). The author raises the question whether \(D(G)\cong T_{\widetilde V}(G)\). In the last part of the paper she deals with the situation where \(G\) is a finite group and the Sylow \(p\)-subgroups of \(G\) are cyclic or of order \(4\).
scientific article; zbMATH DE number 5996443