Tilting equivalence of finite almost derived algebraic cobordism for perfectoid algebras
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913177168510976 |
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| author | Kato, Yuki |
| author_facet | Kato, Yuki |
| contents | We prove tilting equivalence for the finite almost derived algebraic cobordism spectrum $\mathrm{dMGL}^{a,fin}$ of perfectoid algebras. More precisely, for any perfectoid algebra, the tilting functor induces a weak equivalence on the corresponding finite almost derived algebraic cobordism spectra. This invariant is a finite-syntomic, derived, and non-$\msthbb{A}^1$-local version of algebraic cobordism, designed to retain infinitesimal deformation data over mixed-characteristic bases. To prove the result, we isolate a form of excisive approximation for pointed infinity-categories, including non-presentable ones. In the locally finitely presentable case, this agrees with Heuts's framework. We also define approximation functors along natural transformations and apply them to algebraic cobordism and $K$-theory, obtaining results such as Bott periodicity and Gabber rigidity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_10809 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tilting equivalence of finite almost derived algebraic cobordism for perfectoid algebras Kato, Yuki Category Theory 18N55 (primary), 18N70 (secondary) We prove tilting equivalence for the finite almost derived algebraic cobordism spectrum $\mathrm{dMGL}^{a,fin}$ of perfectoid algebras. More precisely, for any perfectoid algebra, the tilting functor induces a weak equivalence on the corresponding finite almost derived algebraic cobordism spectra. This invariant is a finite-syntomic, derived, and non-$\msthbb{A}^1$-local version of algebraic cobordism, designed to retain infinitesimal deformation data over mixed-characteristic bases. To prove the result, we isolate a form of excisive approximation for pointed infinity-categories, including non-presentable ones. In the locally finitely presentable case, this agrees with Heuts's framework. We also define approximation functors along natural transformations and apply them to algebraic cobordism and $K$-theory, obtaining results such as Bott periodicity and Gabber rigidity. |
| title | Tilting equivalence of finite almost derived algebraic cobordism for perfectoid algebras |
| topic | Category Theory 18N55 (primary), 18N70 (secondary) |
| url | https://arxiv.org/abs/2505.10809 |