{"title":"Almost ℂp Galois representations and vector\nbundles","authors":"J. Fontaine","doi":"10.2140/tunis.2020.2.667","DOIUrl":null,"url":null,"abstract":"Let $K$ be a finite extension of $\\mathbb{Q}_p$ and $G_K$ the absolute Galois group. Then $G_K$ acts on the fundamental curve $X$ of $p$-adic Hodge theory and we may consider the abelian category $\\mathcal{M}(G_K)$ of coherent $\\mathcal{O}_X$-modules equipped with a continuous and semi-linear action of $G_K$. An almost $C_p$-representation of $G_K$ is a $p$-adic Banach space $V$ equipped with a linear and continuous action of $G_K$ such that there exists $d\\in\\mathbb{N}$, two $G_K$-stable finite dimensional sub-$\\mathbb{Q}_p$-vector spaces $U_+$ of $V$, $U_-$ of $C_p^d$, and a $G_K$-equivariant isomorphism $V/U_+\\to C_p^d/U_-$. These representations form an abelian category $\\mathcal{C}(G_K)$. The main purpose of this paper is to prove that $\\mathcal{C}(G_K)$ can be recovered from $\\mathcal{M}(G_K)$ by a simple construction (and conversely) inducing, in particular, an equivalence of triangulated categories $D^b(\\mathcal{M}(G_K))\\to D^b(\\mathcal{C}(G_K))$.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2018-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/tunis.2020.2.667","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Tunisian Journal of Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2140/tunis.2020.2.667","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 2
Abstract
Let $K$ be a finite extension of $\mathbb{Q}_p$ and $G_K$ the absolute Galois group. Then $G_K$ acts on the fundamental curve $X$ of $p$-adic Hodge theory and we may consider the abelian category $\mathcal{M}(G_K)$ of coherent $\mathcal{O}_X$-modules equipped with a continuous and semi-linear action of $G_K$. An almost $C_p$-representation of $G_K$ is a $p$-adic Banach space $V$ equipped with a linear and continuous action of $G_K$ such that there exists $d\in\mathbb{N}$, two $G_K$-stable finite dimensional sub-$\mathbb{Q}_p$-vector spaces $U_+$ of $V$, $U_-$ of $C_p^d$, and a $G_K$-equivariant isomorphism $V/U_+\to C_p^d/U_-$. These representations form an abelian category $\mathcal{C}(G_K)$. The main purpose of this paper is to prove that $\mathcal{C}(G_K)$ can be recovered from $\mathcal{M}(G_K)$ by a simple construction (and conversely) inducing, in particular, an equivalence of triangulated categories $D^b(\mathcal{M}(G_K))\to D^b(\mathcal{C}(G_K))$.