The Units of a Ring Spectrum and a Logarithmic Cohomology Operation

Recall that if R is a commutative ring, then the set R× ⊂ R of invertible elements of R is naturally an abelian group under multiplication. This construction is a functor from commutative rings to abelian groups. In general, there is no obvious relation between the additive group of a ring R and the multiplicative group of units R×. However, under certain circumstances one can define a homomorphism from a subgroup of R× to a suitable completion of R, e.g., the natural logarithm Q>0 → R, or the p-adic logarithm (1 + pZp) → Zp. The “logarithmic cohomology operation” is a homotopy-theoretic analogue of the above, where R is a commutative S-algebra and “completion” is Bousfield localization with respect to a Morava K-theory. The purpose of this paper is to give a formula for the logarithmic operation (in certain contexts) in terms of power operations. Before giving our results we briefly explain some of the concepts involved.