The homological slice spectral sequence in motivic and Real bordism

• Published in: arXiv.org 2023-04
• Main Authors: Carrick, Christian, Hill, Michael A, Ravenel, Douglas C
• Format: Article
• Language: English
• Subjects: Convergence; Homology
• ISSN: 2331-8422

For a motivic spectrum \(E\in \mathcal{SH}(k)\), let \(\Gamma(E)\) denote the global sections spectrum, where \(E\) is viewed as a sheaf of spectra on \(\mathrm{Sm}_k\). Voevodsky's slice filtration determines a spectral sequence converging to the homotopy groups of \(\Gamma(E)\). In this paper, we introduce a spectral sequence converging instead to the mod 2 homology of \(\Gamma(E)\) and study the case \(E=BPGL\langle m\rangle\) for \(k=\mathbb R\) in detail. We show that this spectral sequence contains the \(\mathcal{A}_*\)-comodule algebra \((\mathcal{A}//\mathcal{A}(m))^*\) as permanent cycles, and we determine a family of differentials interpolating between \((\mathcal{A}//\mathcal{A}(0))^*\) and \((\mathcal{A}//\mathcal{A}(m))^*\). Using this, we compute the spectral sequence completely for \(m\le 3\). In the height 2 case, the Betti realization of \(BPGL\langle 2\rangle\) is the \(C_2\)-spectrum \(BP_{\mathbb R}\langle 2\rangle\), a form of which was shown by Hill and Meier to be an equivariant model for \(\mathrm{tmf}_1(3)\). Our spectral sequence therefore gives a computation of the comodule algebra \(H_*\mathrm{tmf}_0(3)\). As a consequence, we deduce a new (\(2\)-local) Wood-type splitting \[\mathrm{tmf}\wedge X\simeq \mathrm{tmf}_0(3)\] of \(\mathrm{tmf}\)-modules predicted by Davis and Mahowald, for \(X\) a certain 10-cell complex.