2025/09/23 by Jackson Morris, Morris, Jackson, Sarah C. Petersen +3
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Algebraic structures and combinatorial models #Algebraic Geometry and Number Theory
paper · pdf · doi:10.48550/arxiv.2509.19542
We construct spectrum-level splittings of BPGL ⟨ 1 ⟩ \wedge BPGL ⟨ 1 ⟩ at all primes p, where BPGL ⟨ 1 ⟩ is the first truncated motivic Brown--Peterson spectrum. Classically, BP⟨ 1 ⟩ \wedge BP⟨ 1 ⟩ was first described by Kane and Mahowald in terms of Brown-Gitler spectra. This splitting was subsequently reinterpreted by Lellman and Davis-Gitler-Mahowald in terms of Adams covers. In this paper, we give motivic lifts of these splittings in terms of Adams covers, over the base fields ℂ, ℝ, and \mathbbFq, where \mathbbFq ≠ p. As an application, we compute the E1-page of the BPGL⟨ 1 ⟩-based Adams spectral sequence as a module over BPGL⟨ 1 ⟩, both in homotopy and in terms of motivic spectra. We also record analogous splittings for BPGL ⟨ 0 ⟩ \wedge BPGL ⟨ 0 ⟩.