# Jacobienne locale d'une courbe formelle relative

### Carlos Contou-Carrère

Université de Montpellier II, France

## 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 $S=Speck$ to local symbols of class field theory. Relative local symbols are obtained in terms of the universal property of a couple $(ℑ,f)$, of a $S$-group functor $ℑ$, associated to a $S$-formal curve $X$ locally of the form $X=SpfA[[T]]$ ($S=SpecA)$. $ℑ$ is a $S$-group extension of the completion $Wˇ$ of the universal $S$-Witt vectors group $W$, by the group of units $O_{S}[[T]]_{∗}$. We associate an $S$-functor $ℑ_{omb}$ to $ℑ$, and we define an Abel–Jacobi morphism $f:A=SpecA[[T]][T_{−1}],⟶ℑ_{omb}$ , setting up a group isomorphism:

where $G$ denotes a commutative smooth $S$-group scheme. We define an $S$-bihomomorphism

which is a local symbol (The Tame Symbol), identifying $ℑ$ to its own Cartier dual group $ℑˇ=Hom (ℑ;G_{m})$, and inducing the above isomorphism for $G=G_{m}$. It follows that $ℑ$ may be interpreted as the relative Loop Group:

$S_{′}=SpecA_{′}$ denotes a $S$-scheme, and we write $A_{{S_{′}}}=SpecA_{′}[[T]][T_{−1}]$, and as the $A$-universal group of Witt-Bivectors.

The couple $(ℑ,f)$ may be seen as the local analogue of the relative Rosenlicht Jacobian (Generalized Jacobian) defined by a $S$-smooth curve $X$.

## Cite this article

Carlos Contou-Carrère, Jacobienne locale d'une courbe formelle relative. Rend. Sem. Mat. Univ. Padova 130 (2013), pp. 1–106

DOI 10.4171/RSMUP/130-1