On the cohomology of some Hopf algebroids and Hattori-Stong theorems
The Hattori version of the Stong-Hattori Integrality Theorem holds that the \(KU\)-theory Hurewicz map for the complex bordism spectrum \(MU\), \(MU_*= \pi_*(MU)\to KU_*(MU)\), is a split monomorphism -- that its image is indivisible. The \(KU\)-theory Adams spectral sequence begins with \(\text{Ext}^{*,*}_{KU_* KU}(KU_*,KU*MU)\) and \(\text{Ext}^{0,*}_{KU_* KU}(KU_*, KU*MU) \cong MU_*\). The target of this spectral sequence was ill understood in the late 60's and it was once thought that the Hattori-Stong Theorem was evidence that the spectral sequence collapses. Clarke dispelled this idea by showing that \(\text{Ext}^{1,*}_{KU_*KU}(KU_*,KU*MU)\) is not trivial. When localized at the prime \(p\), \(MU\) splits into copies of the Brown-Peterson spectrum \(BP\). A different splitting reduces localized \(KU_*(-)\) into copies of \(E(1)_*(-)\). This latter theory is the first of a series \(E(n)_*(-)\)'s with coefficients \(E(n)_* \cong \mathbb{Z}_{(p)} [v_1,\dots, v_{n-1}, v_n, v_n^{-1}]\). The author proves analogs of the Hattori-Stong and Clarke results. 1. \(BP_*\to E(n)_*(BP)\) is a split monomorphism. 2. \(\text{Ext}^{s,*}_{E(n)_* E(n)}(E(n)_*,E(n)_*(BP))\) is isomorphic to \[ \begin{cases} BP_* \text{ when }s=0\text{ and to } \\ BP_*/(p^\infty, v_1^\infty, \dots, v_n^\infty)\text{ when }s=n.\end{cases} \] It is trivial for other values of \(s\). These are proved using a chromatic spectral sequence converging to \(\text{Ext}^{s, *}_{E(n)_* E(n)} (E(n)_*,E(n)_*X)\). This is work of Hovey and Sadofsky. The present author reproves the Hovey-Sadofsky results using his own previous work and continuous cohomology of profinite groups.
scientific article; zbMATH DE number 1446310