### 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) |
---|---|

Article number | 106216 |

Journal | Journal of Pure and Applied Algebra |

Volume | 224 |

Issue number | 4 |

DOIs | |

State | Published - Apr 1 2020 |

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### 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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*Journal of Pure and Applied Algebra*, vol. 224, no. 4, 106216. https://doi.org/10.1016/j.jpaa.2019.106216

**A Cartan-Eilenberg spectral sequence for non-normal extensions.** / Belmont, Eva.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A Cartan-Eilenberg spectral sequence for non-normal extensions

AU - Belmont, Eva

PY - 2020/4/1

Y1 - 2020/4/1

N2 - 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.

AB - 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.

KW - Adams spectral sequence

KW - Cartan-Eilenberg spectral sequence

KW - Ext groups

KW - Extension spectral sequence

KW - Hopf algebra cohomology

UR - http://www.scopus.com/inward/record.url?scp=85072220278&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85072220278&partnerID=8YFLogxK

U2 - 10.1016/j.jpaa.2019.106216

DO - 10.1016/j.jpaa.2019.106216

M3 - Article

AN - SCOPUS:85072220278

VL - 224

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

SN - 0022-4049

IS - 4

M1 - 106216

ER -