We consider operations between two multiplicative, complex orientable cohomology theories. Under suitable hypotheses, we construct a map from unstable to stable operations, left-inverse to the usual map from stable to unstable operations. In the main example, where the target theory is one of the Morava K–theories, this provides a simple and explicit description of a splitting arising from the Bousfield–Kuhn functor.
Andrew Stacey. Sarah Whitehouse. "Stable and unstable operations in mod $p$ cohomology theories." Algebr. Geom. Topol. 8 (2) 1059 - 1091, 2008. https://doi.org/10.2140/agt.2008.8.1059