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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2408.02542 |
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| _version_ | 1866929449680764928 |
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| author | Koubaa, Amine |
| author_facet | Koubaa, Amine |
| contents | Let $X$ be a regular scheme over $\textrm{Spec}(\mathbb{Z}[1/p])$ where $p$ is prime. Let $i:Y\to X$ be a closed subscheme of pure codimension $r$. Let $n$ be a natural number prime to $p$. Let $Λ$ be a finite $\mathbb{Z}/n$-module over $X$. In this case, the absolute purity conjectured by Grothendieck and proved by Gabber states that \[
Ri^! Λ\cong Λ_Y(-r)[-2r]\in D_{\textrm{ét}}(Y,Λ) \] For the $n=p^m$-case, a dualizing sheaf was proposed by Milne \cite{MilneValuesOfZeta}, namely the logarithmic de Rham-Witt sheaves $ν_m(r)$. But this doesn't work for all degrees for the étale cohomology. It is however conjectured that this works for the tame cohomology. In this paper we make this work following the proof of Milne in loc. cit. by replacing étale by tame cohomology and assuming resolution of singularities in positive characteristic. We obtain the following isomorphism \[ Ri^! ν_m(n)\cong ν_m(n-r)[-r]. \] |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_02542 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Cartier operator on differentials of discretely ringed adic spaces and Purity in the tame cohomology Koubaa, Amine Algebraic Geometry Let $X$ be a regular scheme over $\textrm{Spec}(\mathbb{Z}[1/p])$ where $p$ is prime. Let $i:Y\to X$ be a closed subscheme of pure codimension $r$. Let $n$ be a natural number prime to $p$. Let $Λ$ be a finite $\mathbb{Z}/n$-module over $X$. In this case, the absolute purity conjectured by Grothendieck and proved by Gabber states that \[ Ri^! Λ\cong Λ_Y(-r)[-2r]\in D_{\textrm{ét}}(Y,Λ) \] For the $n=p^m$-case, a dualizing sheaf was proposed by Milne \cite{MilneValuesOfZeta}, namely the logarithmic de Rham-Witt sheaves $ν_m(r)$. But this doesn't work for all degrees for the étale cohomology. It is however conjectured that this works for the tame cohomology. In this paper we make this work following the proof of Milne in loc. cit. by replacing étale by tame cohomology and assuming resolution of singularities in positive characteristic. We obtain the following isomorphism \[ Ri^! ν_m(n)\cong ν_m(n-r)[-r]. \] |
| title | The Cartier operator on differentials of discretely ringed adic spaces and Purity in the tame cohomology |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2408.02542 |