### 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.

Original language | English (US) |
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Article number | 106216 |

Journal | Journal of Pure and Applied Algebra |

Volume | 224 |

Issue number | 4 |

DOIs | |

State | Published - Apr 1 2020 |

### Keywords

- Adams spectral sequence
- Cartan-Eilenberg spectral sequence
- Ext groups
- Extension spectral sequence
- Hopf algebra cohomology

### ASJC Scopus subject areas

- Algebra and Number Theory