The ring of modules with endo-permutation source. (Q2505563)

Let \(G\) be a finite group, \(p\) be a prime number, \(R\) an algebraically closed field of characteristic \(p>0\) or a suitable \(p\)-adic ring. Moreover, denote by \(a(RG)\) the Green ring of \(RG\) and by \(A(RG)=K\otimes_\mathbb{Z} a(RG)\) the Green algebra of \(RG\), where \(K\) is a suitable field of characteristic \(0\). It is known that the algebra \(A(RG)\) is infinite dimensional for many finite groups, this paper is concerned with some subalgebras of \(A(RG)\). More concrete, let \(A(RG;triv)\) be the subalgebra of \(A(RG)\) spanned by isomorphism classes of indecomposable \(RG\)-modules whose sources are trivial modules, this is finite dimensional and split semisimple. Let \(P\) be a finite \(p\)-group, a finitely generated \(RP\)-module \(M\) is called an `endo-permutation' module if \(\Hom_R(M,M)\cong M\otimes_RM^*\) is a permutation \(RP\)-module, where \(M^*\) denotes the dual of \(M\). Let \(A(RG;ep)\) denote the subalgebra of \(A(RG)\) spanned by isomorphism classes of indecomposable \(RG\)-modules whose sources are endo-permutation modules. On the other hand let \(\mathcal D(RP)\) denote the isomorphism classes of indecomposable endo-permutation \(RP\)-modules with vertex \(P\). It was shown by \textit{L. Puig} [in J. Algebra 131, No. 2, 513-526 (1990; Zbl 0699.20004)] that this is a finitely generated Abelian group. The authors show that \(A(RG;ep)\) is Noetherian and does not contain nilpotent elements. Let \(\mathcal D_t(RP)\) denote the finite subgroup of \(\mathcal D(RP)\) and \(A(RG;tep)\) be the subalgebra spanned by isomorphism classes of indecomposable endo-permutation modules \(M\) with \([M]\in\mathcal D_t(RP)\). The authors show that \(A(RG;tep)\) is a split semisimple \(K\)-algebra and classify the \(K\)-algebra homomorphisms \(A(RG;tep)\to K\). Finally, let \(A(RG;irr)\) be the subalgebra spanned by isomorphism classes of irreducibly generated \(FG\)-modules. The authors show that if \(R=F\) is an algebraically closed field of characteristic \(p>0\) and \(G\) is a \(p\)-solvable finite group then \( A(RG;irr)\) is a split semisimple \(K\)-algebra containing the isomorphism classes of semisimple \(FG\)-modules.