A Cartan-Eilenberg spectral sequence for non-normal extensions

Let Φ→Γ→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 E₂=Ext_Φ(k,Ext_Σ(k,M))⇒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 Φ and Σ. We construct a generalization of the Cartan-Eilenberg spectral sequence converging to Ext_Γ(k,M) that can be defined when Φ=Γ□_Σk is compatibly an algebra and a Γ-comodule; this is related to a construction independently developed by Bruner and Rognes. We show that this spectral sequence is isomorphic, starting at the E₁ 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 E₂ term under an additional flatness assumption. We discuss applications to computing localizations of the Adams spectral sequence E₂ page.