The units of a ring spectrum and a logarithmic cohomology operation

By Charles Rezk. 2004

71 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5fc3702771afffc84fedbf2b12225f539c195f66

Citations in the corpus

• Topological Hochschild homology and cohomology of A∞ ring spectra (2006, unknown): “Recall, eg from [30] that a K.”
• A simple universal property of Thom ring spectra (2014, unknown): “hom category M(f) satisfies a universal property similar to the Thom spectrum. Returning to the example Tζ, this approach is reasonable in view of Clausen’s result [Cla12] saying that Rezk’s logarithm [Rez06] induces an equivalence τ≥1Br(LK(1)S 0) log ∼ //τ ≥1Σ 2L K(1)S 0, at least at odd primes. Con”