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 $X$ 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 ${\Im }_{\text{o}mb}$ 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: