Rendiconti del Seminario Matematico della Università di Padova

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Volume 130, 2013, pp. 1–106
DOI: 10.4171/RSMUP/130-1

Jacobienne locale d'une courbe formelle relative

Carlos Contou-Carrère[1]

(1) Laboratoire Charles Coulomb, UMR 5221, Université de Montpellier II, Case Courrier 051, Place Eugène Bataillon, 34095, Montpellier CEDEX 5, France

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 $(\Im,f)$, of a $S$-group functor $\Im$, associated to a $S$-formal curve ${ \char 88}$ locally of the form ${ \char 88}={\text Spf}\, A&quaa;&quaa;T&quac;&quac;$ ($S=\Spec\,A)$. $\Im$ is a $S$-group extension of the completion $\check{W}$ of the universal $S$-Witt vectors group $W$, by the group of units ${\cal O}_{S}&quaa;&quaa;T&quac;&quac;^{*}$. We associate an $S$-functor $\Im_{\text omb}$ to $\Im$, and we define an Abel-Jacobi morphism $f:{ \char 85}=\Spec\ A&quaa;&quaa;T&quac;&quac;&quaa;T^{-1}&quac;\,\longrightarrow \,\Im_{\text omb}$ , setting up a group isomorphism:

Keywords: Local Symbol(s), Tame Symbol, Contou-Carrere Symbol, Local Abel- Jacobi morphism, Universal Witt Bivectors, Cartier Duality, Local Jacobian, Relative Formal Curve, Witt Residues, Local Relative Class Field Theory, Rosenlicht Jacobian

Contou-Carrère Carlos: Jacobienne locale d'une courbe formelle relative. Rend. Sem. Mat. Univ. Padova 130 (2013), 1-106. doi: 10.4171/RSMUP/130-1