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. These are motivic lifts of Mahowald and Kane's splitting of $BP \langle 1 \rangle \wedge BP \langle 1 \rangle$. These splittings are given in terms of motivic Adams covers and constructed over the base fields $\mC, \, \mR,$ and $\mF_q$, where $\textup{char}(\mathbb{F}_q) \neq p$. 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 $BPGL \langle 0 \rangle \wedge BPGL \langle 0 \rangle$.