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| Natura: | Preprint |
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2026
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| Accesso online: | https://arxiv.org/abs/2605.06515 |
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| _version_ | 1866913103464103936 |
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| author | Balderrama, William Davies, Jack Morgan Linskens, Sil |
| author_facet | Balderrama, William Davies, Jack Morgan Linskens, Sil |
| contents | Given a global equivariant ultracommutative ring spectrum $E$ and inclusion $H\hookrightarrow G$ of finite groups, one may apply geometric fixed points to the norm $N_H^G E_H \to E_G$ to obtain what we call a \emph{geometric norm} $Φ^H E \to Φ^G E$. We prove that, together with inflations, these assemble into a functor $Φ\colon\mathrm{UCom}_{\mathrm{fin}} \to \mathrm{Fun}(\mathrm{Span}(\mathcal{G},\mathcal{E},\mathcal{O}),\mathrm{CAlg})$, where $\mathrm{Span}(\mathcal{G},\mathcal{E},\mathcal{O})$ is the span category of finite connected groupoids with full backwards maps and faithful forwards maps, and that $Φ$ restricts to an equivalence between full subcategories of rational objects.
Central to our construction is a refinement of geometric fixed points to a natural transformation $Φ\colon \mathrm{Sp}_\bullet\to\mathrm{Fun}(\mathrm{Orb}_\bullet^\simeq,\mathrm{Sp})$ which is compatible with restrictions and norms, and which restricts to an equivalence on full subcategories of rational objects. We explain how this may also be used to recover theorems of Barrero--Barthel--Pol--Strickland--Williamson and Wimmer on algebraic models for rational global spectra and normed $G$-commutative ring spectra respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_06515 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An algebraic model for rational ultracommutative rings Balderrama, William Davies, Jack Morgan Linskens, Sil Algebraic Topology 55P91 Given a global equivariant ultracommutative ring spectrum $E$ and inclusion $H\hookrightarrow G$ of finite groups, one may apply geometric fixed points to the norm $N_H^G E_H \to E_G$ to obtain what we call a \emph{geometric norm} $Φ^H E \to Φ^G E$. We prove that, together with inflations, these assemble into a functor $Φ\colon\mathrm{UCom}_{\mathrm{fin}} \to \mathrm{Fun}(\mathrm{Span}(\mathcal{G},\mathcal{E},\mathcal{O}),\mathrm{CAlg})$, where $\mathrm{Span}(\mathcal{G},\mathcal{E},\mathcal{O})$ is the span category of finite connected groupoids with full backwards maps and faithful forwards maps, and that $Φ$ restricts to an equivalence between full subcategories of rational objects. Central to our construction is a refinement of geometric fixed points to a natural transformation $Φ\colon \mathrm{Sp}_\bullet\to\mathrm{Fun}(\mathrm{Orb}_\bullet^\simeq,\mathrm{Sp})$ which is compatible with restrictions and norms, and which restricts to an equivalence on full subcategories of rational objects. We explain how this may also be used to recover theorems of Barrero--Barthel--Pol--Strickland--Williamson and Wimmer on algebraic models for rational global spectra and normed $G$-commutative ring spectra respectively. |
| title | An algebraic model for rational ultracommutative rings |
| topic | Algebraic Topology 55P91 |
| url | https://arxiv.org/abs/2605.06515 |