{"title":"Cohomologie des courbes analytiques $p$-adiques","authors":"P. Colmez, Gabriel Dospinescu, Wiesława Nizioł","doi":"10.4310/cjm.2022.v10.n3.a1","DOIUrl":null,"url":null,"abstract":"Cohomology of affinoids does not behave well; often, this can be remedied by making affinoids overconvergent. In this paper, we focus on dimension 1 and compute, using analogs of pants decompositions of Riemann surfaces, various cohomologies of affinoids. To give a meaning to these decompositions we modify slightly the notion of $p$-adic formal scheme, which gives rise to the adoc (an interpolation between adic and ad hoc) geometry. It turns out that cohomology of affinoids (in dimension 1) is not that pathological. From this we deduce a computation of cohomologies of curves without boundary (like the Drinfeld half-plane and its coverings). In particular, we obtain a description of their $p$-adic pro-\\'etale cohomology in terms of de the Rham complex and the Hyodo-Kato cohomology, the later having properties similar to the ones of $\\ell$-adic pro-\\'etale cohomology, for $\\ell\\neq p$.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.8000,"publicationDate":"2021-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Cambridge Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4310/cjm.2022.v10.n3.a1","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 2
Abstract
Cohomology of affinoids does not behave well; often, this can be remedied by making affinoids overconvergent. In this paper, we focus on dimension 1 and compute, using analogs of pants decompositions of Riemann surfaces, various cohomologies of affinoids. To give a meaning to these decompositions we modify slightly the notion of $p$-adic formal scheme, which gives rise to the adoc (an interpolation between adic and ad hoc) geometry. It turns out that cohomology of affinoids (in dimension 1) is not that pathological. From this we deduce a computation of cohomologies of curves without boundary (like the Drinfeld half-plane and its coverings). In particular, we obtain a description of their $p$-adic pro-\'etale cohomology in terms of de the Rham complex and the Hyodo-Kato cohomology, the later having properties similar to the ones of $\ell$-adic pro-\'etale cohomology, for $\ell\neq p$.
仿射的上同性表现不佳;通常,这可以通过使仿射过度收敛来补救。在本文中,我们将重点放在维1上,并使用类似于黎曼曲面的分解,计算各种仿射的上同调。为了给这些分解赋予意义,我们稍微修改了$p$-adic形式方案的概念,这就产生了adoc(在adic和ad hoc之间的插值)几何。事实证明,仿射的上同调(在1维)并不是那么病态。由此我们推导出无边界曲线(如德林菲尔德半平面及其覆盖)的上同调的计算。特别地,我们用Rham复形和Hyodo-Kato上同调来描述它们的$p$-adic上同调,后者对于$\ well \neq p$具有类似于$\ well $-adic上同调的性质。