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Jacobienne locale d'une courbe formelle relative

Résumé : This article is devoted to the proof of a relative duality formula on a noetherian scheme $S$, giving rise on the spectrum of a field $S=Spec\,k$ to local symbols of class field theory. Relative local symbols are obtained in terms of the universal property of a couple $(\mathfrak{J},f)$, of a $S$-group functor $\mathfrak{J}$, associated to a $S$-formal curve $\mathfrak{X}$ locally of the form $\mathfrak{X}=Spf\, A[[T]]$ ($S=Spec\,A)$. $\mathfrak{J}$ is a $S$-group extension of the completion $\check{W}$ of the universal $S$-Witt vectors group $W$, by the group of units $\mathcal{O}_{S}[[T]]^{*}$. We associate an $S$-functor $\mathfrak{J}_{omb}$ to $\mathfrak{J}$, and we define an Abel-Jacobi morphism $f:\mathfrak{U}=Spec\ A[[T]][T^{-1}]\longrightarrow \mathfrak{J}_{omb}$ , setting up a group isomorphism: $$Hom_{S-gr}(\mathfrak{J},G)\simeq G(\mathfrak{U}),$$ where $G$ denotes a commutative smooth $S$-group scheme. We define an $S$-bihomomorphism $$\mathfrak{J}\times\mathfrak{J}\longrightarrow\mathbb{G}_{m},$$ which is a local symbol (The Tame Symbol), identifying $\mathfrak{J}$ to its own Cartier dual group $\check{\mathfrak{J}}=\underline{Hom}_{S-gr}(\mathfrak{J},\mathbb{G}_{m})$, and inducing the above isomorphism for $G=\mathbb{G}_{m}$. It follows that $\mathfrak{J}$ may be interpreted as the relative Loop Group: $$\underline{\mathbb{G}}_{m}(\mathfrak{U}):S'\longrightarrow G(\mathfrak{U}_{\{S'\}})\ ,$$ $S'=Spec\ A'$ denotes a $S$-scheme, and we write $\mathfrak{U}_{\{S'\}}=Spec\ A'[[T]][T^{-1}]$, and as the $A$-universal group of Witt-Bivectors.\\ The couple $(\mathfrak{J},f)$ may be seen as the local analogue of the relative Rosenlicht Jacobian (Generalized Jacobian) defined by a $S$-smooth curve $X$.
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Carlos Contou-Carrere. Jacobienne locale d'une courbe formelle relative. Rendiconti del Seminario Matematico della Università di Padova, University of Padua / European Mathematical Society, 2013, 130, pp.1-106. ⟨10.4171/RSMUP/130-1⟩. ⟨hal-01286493⟩

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