Jacobienne locale d'une courbe formelle relative

Abstract

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: