Splittings of truncated motivic Brown--Peterson cooperations algebras

Abstract

We construct spectrum-level splittings of $BPGL \langle 1 \rangle \wedge BPGL \langle 1 \rangle$ at all primes $p$, where $BPGL \langle 1 \rangle$ is the first truncated motivic Brown--Peterson spectrum. Classically, $BP\langle 1 \rangle \wedge BP\langle 1 \rangle$ 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 $\mathbb{F}_q \neq p$. As an application, we compute the $E_1$-page of the $BPGL\langle 1 \rangle$-based Adams spectral sequence as a module over $BPGL\langle 1 \rangle$, both in homotopy and in terms of motivic spectra. We also record analogous splittings for $BPGL \langle 0 \rangle \wedge BPGL \langle 0 \rangle$.