Cohomology of the Morava stabilizer group through the duality resolution at \(n=p=2\) (Q6567120)

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 the range \(0\leq t <12\), also using input from [\textit{A. Beaudry} et al., Geom. Topol. 26, No. 1, 377--476 (2022; Zbl 1494.55016)]. (The authors indicate that, to generalize to all \(t\), one would require to control the differential \(d_1\) in the ADSS.)\N\NPassage to \(H^* (\mathbb{S}_2; \mathbf{E}_*)\) is achieved by using the Serre spectral sequence for \(\mathbb{S}_2 \cong \mathbb{S}^1_2 \rtimes \mathbb{Z}_2\). In their range, again using input from previous work, they show that the \(\mathbb{Z}_2\)-action on \(H^* (\mathbb{S}^1_2; \mathbf{E}_*)\) is trivial and that there are no additive extensions.\N\NPassage to \(H^* (\mathbb{G}^1_2 ; \mathbf{E}_*)\) and \(H^* (\mathbb{G}_2 ; \mathbf{E}_*)\) is based on the identifications \(H^* (\mathbb{S}^1_2 ; \mathbf{E}_*)\cong \mathbb{W} \otimes_{\mathbb{Z}_2} H^* (\mathbb{G}^1_2 ; \mathbf{E}_*)\) and \(H^* (\mathbb{S}_2 ; \mathbf{E}_*)\cong \mathbb{W} \otimes_{\mathbb{Z}_2} H^* (\mathbb{G}_2 ; \mathbf{E}_*)\), where \(\mathbb{W}\) denotes the Witt vectors. To facilitate this, the authors made judicious choices of Galois-invariant generators along the way.\N\NThe analysis of the \(d_3\)-differentials in the HFPSS exploits the known behaviour of the \(v_1\)-localized \(K(2)\)-local \(\mathbf{E}\)-based Adams-Novikov spectral sequence.