Splittings of truncated motivic Brown--Peterson cooperations algebras
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918147818258432 |
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| author | Morris, Jackson Petersen, Sarah Tatum, Elizabeth |
| author_facet | Morris, Jackson Petersen, Sarah Tatum, Elizabeth |
| contents | 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 $BPGL \langle 0 \rangle \wedge BPGL \langle 0 \rangle$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_19542 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Splittings of truncated motivic Brown--Peterson cooperations algebras Morris, Jackson Petersen, Sarah Tatum, Elizabeth Algebraic Topology K-Theory and Homology 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 $BPGL \langle 0 \rangle \wedge BPGL \langle 0 \rangle$. |
| title | Splittings of truncated motivic Brown--Peterson cooperations algebras |
| topic | Algebraic Topology K-Theory and Homology |
| url | https://arxiv.org/abs/2509.19542 |