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

Abstract

Let $\Phi\to \Gamma\to \Sigma$ be a conormal extension of Hopf algebras over a commutative ring $k$, and let $M$ be a $\Gamma$-comodule. The Cartan-Eilenberg spectral sequence $$ E_2 = \mathrm{Ext}_\Phi(k,\mathrm{Ext}_\Sigma(k,M)) \implies \mathrm{Ext}_\Gamma(k,M)$$ is a standard tool for computing the Hopf algebra cohomology of $\Gamma$ with coefficients in $M$ in terms of the cohomology of the pieces $\Phi$ and $\Sigma$. Bruner and Rognes, generalizing a construction of Davis and Mahowald, have introduced a generalization of the Cartan-Eilenberg spectral sequence converging to $\mathrm{Ext}_\Gamma(k,M)$ that can be defined when $\Phi = \Gamma\square_\Sigma k$ is compatibly an algebra and a $\Gamma$-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 $\Gamma$-comodules as studied by Margolis and Palmieri, and to a filtration spectral sequence on the cobar complex for $\Gamma$ 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.