Commentarii Mathematici Helvetici

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Volume 80, Issue 4, 2005, pp. 691–723
DOI: 10.4171/CMH/31

Published online: 2005-12-31

On the $\Gamma$-cohomology of rings of numerical polynomials

Andrew Baker[1] and Birgit Richter

(1) University of Glasgow, UK

We investigate $\Gamma$-cohomology of some commutative cooperation algebras $E_*E$ associated with certain periodic cohomology theories. For KU and $E(1)$, the Adams summand at a prime $p$, and for KO we show that $\Gamma$-cohomology vanishes above degree 1. As these cohomology groups are the obstruction groups in the obstruction theory developed by Alan Robinson we deduce that these spectra admit unique $E_\infty$ structures. As a consequence we obtain an $E_\infty$ structure for the connective Adams summand. For the Johnson--Wilson spectrum $E(n)$ with $n\geq1$ we establish the existence of a unique $E_\infty$ structure for its $I_n$-adic completion.

Keywords: Structured ring spectra, $\Gamma$-cohomology, $K$-theory, Johnson--Wilson spectra

Baker Andrew, Richter Birgit: On the $\Gamma$-cohomology of rings of numerical polynomials. Comment. Math. Helv. 80 (2005), 691-723. doi: 10.4171/CMH/31