Operations on ring structures preserved by normalized automorphisms of group rings
Let \(G\) be a finite group and \(R\) a commutative ring. The authors study the action of an automorphism \(\alpha\) of \(RG\) preserving the augmentation. Suppose that \(R\) is an integral domain of characteristic \(0\) such that no prime divisor of the order of \(G\) is invertible in \(R\). Then, the authors prove that \(\alpha\) commutes with the \(\lambda\)-ring structure of the Grothendieck group \(G_0(kG)\) for \(k\) being any field. The method is to elaborate the relation between the \(\lambda\)-ring structure and the Adams operations. Let now \(R\) be the ring of integers in a number field. Then, the authors use this result to prove that \(\alpha(X\cdot M)=\alpha(X)\cdot\alpha(M)\) where \(X\) is an element of the Grothendieck group \(G^R_0(RG)\) of \(RG\)-lattices and \(M\) is an object in one of the following \(G^R_0(RG)\)-modules: the Grothendieck group of finitely generated projective \(RG\)-modules, the locally free class group, the kernel subgroup, the torsion subgroup of the Grothendieck group of finitely generated \(R\)-modules. The method is to exhibit Fröhlich's Hom-description of class groups as well as Queyrut's Hom-description for the torsion part of the Grothendieck group of finitely generated \(RG\)-modules.
scientific article; zbMATH DE number 1309127
scientific article