Gersten conjecture for K-theory on Henselian schemes and $ϕ$-motivic localisation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908684474384384 |
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| author | Druzhinin, Andrei E |
| author_facet | Druzhinin, Andrei E |
| contents | A key triviality result for support extension maps for motivic $\mathbb{A}^1$-homotopies of cellular motivic spaces $S$ over a DVR spectrum $B$ is proven. Combining with earlier known results on Gersten complex and the K-theory motivic spectrum we achieve a proof of the Gersten Conjecture for K-theory on essentially smooth local Henselian $B$-schemes. Additionally, we outline generalisations for Cousin complexes associated to motivic $\mathbb{A}^1$- and $\square$-homotopies of cellular $B$-spectra.
The proof is based on two ingredients:
(1) A new ``motivic localisation'' over $B$, called \emph{$ϕ$-motivic}, % localisation giving rise to the $ϕ$-motivic homotopy category such that the triviality of the support extension maps and the acyclicity of Cousin complexes hold for all objects $S$, not necessarily cellular.
(2) An interpretation of some classes in the motivic $\mathbb{A}^1$-homotopies with support defined with respect to the Morel-Voevodsky motivic homotopy category of smooth $B$-schemes in terms of the construction of $ϕ$-motivic homotopy category mentioned in Point (1). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_01923 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Gersten conjecture for K-theory on Henselian schemes and $ϕ$-motivic localisation Druzhinin, Andrei E K-Theory and Homology 19D99, 14F99 A key triviality result for support extension maps for motivic $\mathbb{A}^1$-homotopies of cellular motivic spaces $S$ over a DVR spectrum $B$ is proven. Combining with earlier known results on Gersten complex and the K-theory motivic spectrum we achieve a proof of the Gersten Conjecture for K-theory on essentially smooth local Henselian $B$-schemes. Additionally, we outline generalisations for Cousin complexes associated to motivic $\mathbb{A}^1$- and $\square$-homotopies of cellular $B$-spectra. The proof is based on two ingredients: (1) A new ``motivic localisation'' over $B$, called \emph{$ϕ$-motivic}, % localisation giving rise to the $ϕ$-motivic homotopy category such that the triviality of the support extension maps and the acyclicity of Cousin complexes hold for all objects $S$, not necessarily cellular. (2) An interpretation of some classes in the motivic $\mathbb{A}^1$-homotopies with support defined with respect to the Morel-Voevodsky motivic homotopy category of smooth $B$-schemes in terms of the construction of $ϕ$-motivic homotopy category mentioned in Point (1). |
| title | Gersten conjecture for K-theory on Henselian schemes and $ϕ$-motivic localisation |
| topic | K-Theory and Homology 19D99, 14F99 |
| url | https://arxiv.org/abs/2512.01923 |