Atiyah duality for motivic spectra
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arXiv
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| Format: | Preprint |
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2024
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| author | Annala, Toni Hoyois, Marc Iwasa, Ryomei |
| author_facet | Annala, Toni Hoyois, Marc Iwasa, Ryomei |
| contents | We prove that Atiyah duality holds in the $\infty$-category of non-$\mathbb A^1$-invariant motivic spectra over arbitrary derived schemes: every smooth projective scheme is dualizable with dual given by the Thom spectrum of its negative tangent bundle. The Gysin maps recently constructed by L. Tang are a key ingredient in the proof. We then present several applications. First, we study $\mathbb A^1$-colocalization, which transforms any module over the $\mathbb A^1$-invariant sphere into an $\mathbb A^1$-invariant motivic spectrum without changing its values on smooth projective schemes. This can be applied to all known $p$-adic cohomology theories and gives a new elementary approach to "logarithmic" or "tame" cohomology theories; it recovers for instance the logarithmic crystalline cohomology of strict normal crossings compactifications over perfect fields and shows that the latter is independent of the choice of compactification. Second, we prove a motivic Landweber exact functor theorem, associating a motivic spectrum to any graded formal group law classified by a flat map to the moduli stack of formal groups. Using this theorem, we compute the ring of $\mathbb P^1$-stable cohomology operations on the algebraic K-theory of qcqs derived schemes, and we prove that rational motivic cohomology is an idempotent motivic spectrum. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_01561 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Atiyah duality for motivic spectra Annala, Toni Hoyois, Marc Iwasa, Ryomei Algebraic Geometry Algebraic Topology K-Theory and Homology We prove that Atiyah duality holds in the $\infty$-category of non-$\mathbb A^1$-invariant motivic spectra over arbitrary derived schemes: every smooth projective scheme is dualizable with dual given by the Thom spectrum of its negative tangent bundle. The Gysin maps recently constructed by L. Tang are a key ingredient in the proof. We then present several applications. First, we study $\mathbb A^1$-colocalization, which transforms any module over the $\mathbb A^1$-invariant sphere into an $\mathbb A^1$-invariant motivic spectrum without changing its values on smooth projective schemes. This can be applied to all known $p$-adic cohomology theories and gives a new elementary approach to "logarithmic" or "tame" cohomology theories; it recovers for instance the logarithmic crystalline cohomology of strict normal crossings compactifications over perfect fields and shows that the latter is independent of the choice of compactification. Second, we prove a motivic Landweber exact functor theorem, associating a motivic spectrum to any graded formal group law classified by a flat map to the moduli stack of formal groups. Using this theorem, we compute the ring of $\mathbb P^1$-stable cohomology operations on the algebraic K-theory of qcqs derived schemes, and we prove that rational motivic cohomology is an idempotent motivic spectrum. |
| title | Atiyah duality for motivic spectra |
| topic | Algebraic Geometry Algebraic Topology K-Theory and Homology |
| url | https://arxiv.org/abs/2403.01561 |