Adams operations and algebraic \(K\)-theory
Let \(X\) be a scheme furnished with a big family of invertible bundles, the author shows in this paper that the graded ring \[ \bigoplus_{i\geq 0}K_i(X)\otimes\mathbb Q \] adquires a structure of \(\lambda\)-ring from the Schechtman operations \[ \Psi_k:K_i(X)\otimes\mathbb Q\rightarrow K_i(X)\otimes\mathbb Q \] for \(i\geq 0\) in Quillen's \(K\)-theory. After a good exposition of bisimplicial schemes, he defines the \(K\)-theory groups of multisimplicial schemes, and for every group \(G\) a morphism \[ R_\mathbb Z(G)\rightarrow K_0(BG) \] where \(R_\mathbb Z\) stands for the Grothendieck ring of the algebraic representations of \(G\), and he computes some particular images for the case \(G=GL(n)\). It is also shown that the group \(K_{-1}(N{\mathcal U}\otimes{\mathbb B}P(N{\mathcal U}))\) is isomorphic to the group \([|{\mathbb B}P(N{\mathcal U})|,K(N{\mathcal U})]\), and finally, that the identities of \(\lambda\)-rings are transported from \(R_\mathbb Z(GL(n))\) to \(K_i(X)\otimes\mathbb Q\).
scientific article; zbMATH DE number 1132931