Cohomology of the Morava stabilizer group through the duality resolution at \(n=p=2\)
type: Item
instance of: scholarly article
title: Cohomology of the Morava stabilizer group through the duality resolution at \(n=p=2\)
author: Agnès Beaudry
author: Irina Bobkova
author: Paul Goerss
author: Hans-Werner Henn
author: Viet-Cuong Pham
author: Vesna Stojanoska
published in: Transactions of the American Mathematical Society
publication date: 2024-07-04T00:00:00Z
review text: This paper is motivated by the study of \(K(2)\)-local stable homotopy theory at the prime \(p=2\), in particular that of the \(K(2)\)-local sphere spectrum \(L_{K(2)}S^0\) via the homotopy fixed point spectral sequence (HFPSS)\N\[\NH^* (\mathbb{G}_2; \mathbf{E}_*) \Rightarrow \pi_* L_{K(2)} S^0.\N\]\NAs usual, the authors take \(\Gamma\) to be the formal group of the super singular elliptic curve over \(\mathbb{F}_4\) and work with the profinite groups \(\mathbb{S}_2:= \mathrm{Aut}(\Gamma)\) and \(\mathbb{G}_2:= \mathbb{S}_2 \rtimes \mathrm{Gal}(\mathbb{F}_4/\mathbb{F}_2)\); the subgroup \(\mathbb{S}^1_2\) is the kernel of \(\zeta :\mathbb{S}_2 \twoheadrightarrow \mathbb{Z}_2\) induced by the determinant and \(\mathbb{G}^1_2\subset \mathbb{G}_2\) is defined analogously. The ring spectrum \(\mathbf{E}:= \mathbf{E} (\mathbb{F}_4, \Gamma)\) is the associated height two Lubin-Tate theory and \(\mathbf{E}_t := \pi_t \mathbf{E}\).\N\NTheir main results determine \(H^* (\mathbb{G}_2; \mathbf{E}_t)\) for \(0 \leq t <12\) and the differential \(d_3\) in the HFPSS in this range. These are impressive, intricate calculations, building upon previous results.\N\NThe first steps are accomplished by using the algebraic duality spectral sequence (ADSS) based on the algebraic duality resolution of \(\mathbb{S}^1_2\)-modules\N\[\N0 \rightarrow \mathcal{C}_3 \rightarrow \mathcal{C}_2 \rightarrow \mathcal{C}_1 \rightarrow \mathcal{C}_0 \rightarrow \mathbb{Z}_2 \rightarrow 0,\N\]\Nwhere \(\mathcal{C}_i = \mathbb{Z}_2 [[\mathbb{S}^1_2/F_i]]\) with \(F_0= G_{24}\), \(F_1 = F_2 = C_6\), and \(F_3 = G'_{24}\), finite subgroups of \(\mathbb{S}^1_2\). The \(E_1\)-page is expressed in terms of the cohomology of these finite groups that is recalled in the paper. The ADSS was used in [\textit{A. Beaudry}, Adv. Math. 306, 722--788 (2017; Zbl 1420.55031)] to understand \(H^* (\mathbb{S}^1_2; \mathbf{E}_*/2)\); her results are bootstrapped here to calculate \(H^* (\mathbb{S}^1_2; \mathbf{E}_*)\) in…
reviewed by: Geoffrey Powell
Mathematics Subject Classification ID: 55P42
zbMATH DE Number: 7876004
zbMATH Keywords: algebraic duality spectral sequence
zbMATH Keywords: homotopy fixed point spectral sequence
zbMATH Keywords: \(K(2)\)-local sphere
zbMATH Keywords: chromatic homotopy theory
MaRDI profile type: Publication
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DOI: 10.1090/TRAN/8981
description: scientific article; zbMATH DE number 7876004