2018/11/13 by Eva Belmont, Belmont, Eva
Mathematics · #18G40 #55T15 #57T05 #57T30 #Algebraic Topology (math.AT) #FOS: Mathematics #math.AT #msc:18G40 #msc:55T15 #msc:57T05 #msc:57T30
paper · pdf · doi:10.48550/arxiv.1811.05459
Acknowledged relevant work of Davis-Mahowald and Bruner-Rognes
arxiv created 2019/01/19 · arxiv updated 2019/01/23
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 the pieces Φ and Σ. Bruner and Rognes, generalizing a construction of Davis and Mahowald, have introduced a generalization of the Cartan-Eilenberg spectral sequence converging to ExtΓ(k,M) that can be defined when Φ= Γ\squareΣk is compatibly an algebra and a Γ-comodule. We offer a concrete cobar-like construction that fits into their framework, and show how this work fits into a larger story. In particular, 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.