A Cartan-Eilenberg spectral sequence for non-normal extensions

Eva Belmont

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


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=ExtΦ(k,ExtΣ(k,M))⇒ExtΓ(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 ExtΓ(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.

Original languageEnglish (US)
Article number106216
JournalJournal of Pure and Applied Algebra
Issue number4
StatePublished - Apr 1 2020


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

ASJC Scopus subject areas

  • Algebra and Number Theory


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