A Cartan-Eilenberg spectral sequence for non-normal extensions

Abstract 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 E 2 = 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 □ Σ k 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 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.