On \(\lambda\)-rings and the \(K\)-theory of infinite loop spaces
The notion of \(\mathbb{Z}/2\)-graded \(\lambda\)-ring \(R\) is introduced; \(R^0\) is a \(\lambda\)-ring, while \(R^1\) is a \(\psi\)-module. Next, the simpler notion of \(\mathbb{Z}/2\)-graded \(\theta^p\)-ring is introduced; it just requires functions \(\theta^p: R^0\to R^0\) and \(\psi^p: R^1\to R^1\) satisfying certain properties. It is proved that \(p\)-local \(\mathbb{Z}/2\)-graded \(\lambda\)-rings are equivalent to \(p\)-local \(\mathbb{Z}/2\)-graded \(\theta^p\)-rings equipped with Adams operations \(\psi^k\) for all \(k\) prime to \(p\). This is then extended to \(p\)-adic rings. The main topological result gives the \(\mathbb{Z}/2\)-graded \(p\)-adic \(\lambda\)-ring \(K^*(\Omega^\infty E; \mathbb{Z}_p)\) and Hopf algebra \(K_*(\Omega^\infty E; \mathbb{Z}/p)\) for a spectrum \(E\) with \(K^*(E; \mathbb{Z}_p)\) torsion-free. If \(E\) is 1-connected with \(\pi_2 E\) torsion, then \(K^*(\Omega^\infty E; \mathbb{Z}_p)\) is the connective \(p\)-adic \(\theta^p\)-ring freely generated by \(K^*(E; \mathbb{Z}_p)\), and \(K_*(\Omega^\infty E; \mathbb{Z}/p)\) is a tensor product of Witt-Hopf algebras and exterior Hopf algebras.
scientific article; zbMATH DE number 847542
scientific article