A Cartan-Eilenberg spectral sequence for a non-normal extension

Abstract

Let $Φ\to Γ\to Σ$ be a conormal extension of Hopf algebras over a commutative ring $k$, and let $M$ be a $Γ$-comodule. The Cartan-Eilenberg spectral sequence $$ E_2 = \mathrm{Ext}_Φ(k,\mathrm{Ext}_Σ(k,M)) \implies \mathrm{Ext}_Γ(k,M)$$ is a standard tool for computing the Hopf algebra cohomology of $Γ$ with coefficients in $M$ in terms of the cohomology of the pieces $Φ$ and $Σ$. Bruner and Rognes, generalizing a construction of Davis and Mahowald, have introduced a generalization of the Cartan-Eilenberg spectral sequence converging to $\mathrm{Ext}_Γ(k,M)$ that can be defined when $Φ= Γ\square_Σk$ is compatibly an algebra and a $Γ$-comodule. We offer a concrete cobar-like construction that fits into their framework, and show how this work fits into a larger story. In particular, we show that this spectral sequence is isomorphic, starting at the $E_1$ page, to both the Adams spectral sequence in the stable category of $Γ$-comodules as studied by Margolis and Palmieri, and to a filtration spectral sequence on the cobar complex for $Γ$ originally due to Adams. We obtain a description of the $E_2$ term under an additional flatness assumption. We discuss applications to computing localizations of the Adams spectral sequence $E_2$ page.