A total Steenrod operation as homomorphism of Steenrod algebra-modules

摘要

Let p be an odd prime, M be an unstable locally finite module over the Steenrod algebra, and let ${Phi_{S}}$ be the localization out of the Euler class of the mod-p cohomology ring of the group Z/p. We prove that the Singer evaluation map ${d : Phi_S^{GL_{1}} otimes M longrightarrow M }$ is dually related to a total operation ${chi P : M longrightarrow Phi_S^{GL_{1}} otimes M}$ . We determine the exotic ${mathcal A (p)}$ -module structure on the target which makes χ P an ${mathcal A (p)}$ -linear map, give a new proof of the Adem relations and find some new identities involving the Bockstein and the pth reduced powers.