A Cartan-Eilenberg spectral sequence for non-normal extensions

Research output: Contribution to journalArticle

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 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
Volume224
Issue number4
DOIs
StatePublished - Apr 1 2020

Fingerprint

Spectral Sequence
Comodule
Adams Spectral Sequence
Hopf Algebra
Cohomology
Computing
Flatness
Commutative Ring
Filtration
Isomorphic
Algebra
Coefficient
Term

Keywords

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

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

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title = "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 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.",
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A Cartan-Eilenberg spectral sequence for non-normal extensions. / Belmont, Eva.

In: Journal of Pure and Applied Algebra, Vol. 224, No. 4, 106216, 01.04.2020.

Research output: Contribution to journalArticle

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