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:
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