Splittings of truncated motivic Brown--Peterson cooperations algebras

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 $\mathbb{C},\mathbb{R},$ and ${\mathbb{F}}_q$, where ${\mathbb{F}}_q\mathrm{\ne }p$. As an application, we compute the $E_1$-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 $BPGL⟨0⟩\wedge BPGL⟨0⟩$.