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

TL;DR

This paper develops a generalized Cartan-Eilenberg spectral sequence for non-normal extensions of Hopf algebras, connecting it to Adams spectral sequences and providing tools for computing Hopf algebra cohomology.

Contribution

It introduces a concrete cobar-like construction for the generalized spectral sequence and relates it to existing spectral sequences in stable homotopy theory.

Findings

The spectral sequence is isomorphic to the Adams spectral sequence starting at E_1.

Provides a description of the E_2 term under flatness assumptions.

Applications to localizations of the Adams spectral sequence E_2 page.

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.