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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Online-Zugang: | https://arxiv.org/abs/2410.16757 |
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| _version_ | 1866917812536082432 |
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| author | Hoyois, Marc |
| author_facet | Hoyois, Marc |
| contents | We generalize several basic facts about the motivic sphere spectrum in $\mathbb A^1$-homotopy theory to the category $\mathrm{MS}$ of non-$\mathbb A^1$-invariant motivic spectra over a derived scheme. On the one hand, we show that all the Milnor-Witt K-theory relations hold in the graded endomorphism ring of the motivic sphere. On the other hand, we show that the positive eigenspace $\mathbf 1_\mathbb Q^+$ of the rational motivic sphere is the rational motivic cohomology spectrum $\mathrm H\mathbb Q$, which represents the eigenspaces of the Adams operations on rational algebraic K-theory. We deduce several familiar characterizations of $\mathrm H\mathbb Q$-modules in $\mathrm{MS}$: a rational motivic spectrum is an $\mathrm H\mathbb Q$-module iff it is orientable, iff the involution $\langle -1\rangle$ is the identity, iff the Hopf map $η$ is zero, iff it satisfies étale descent. Moreover, these conditions are automatic in many cases, for example over non-orderable fields and over $\mathbb Z[ζ_n]$ for any $n\geq 3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_16757 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Remarks on the motivic sphere without $\mathbb A^1$-invariance Hoyois, Marc Algebraic Geometry Algebraic Topology K-Theory and Homology We generalize several basic facts about the motivic sphere spectrum in $\mathbb A^1$-homotopy theory to the category $\mathrm{MS}$ of non-$\mathbb A^1$-invariant motivic spectra over a derived scheme. On the one hand, we show that all the Milnor-Witt K-theory relations hold in the graded endomorphism ring of the motivic sphere. On the other hand, we show that the positive eigenspace $\mathbf 1_\mathbb Q^+$ of the rational motivic sphere is the rational motivic cohomology spectrum $\mathrm H\mathbb Q$, which represents the eigenspaces of the Adams operations on rational algebraic K-theory. We deduce several familiar characterizations of $\mathrm H\mathbb Q$-modules in $\mathrm{MS}$: a rational motivic spectrum is an $\mathrm H\mathbb Q$-module iff it is orientable, iff the involution $\langle -1\rangle$ is the identity, iff the Hopf map $η$ is zero, iff it satisfies étale descent. Moreover, these conditions are automatic in many cases, for example over non-orderable fields and over $\mathbb Z[ζ_n]$ for any $n\geq 3$. |
| title | Remarks on the motivic sphere without $\mathbb A^1$-invariance |
| topic | Algebraic Geometry Algebraic Topology K-Theory and Homology |
| url | https://arxiv.org/abs/2410.16757 |