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 $S=\mathrm{Spec}k$ 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 $\mathfrak{X}$ locally of the form $\mathfrak{X}=\mathrm{Spf}A[[T]]$ ($S=\mathrm{Spec}A)$. $\Im$ is a $S$-group extension of the completion $\check{W}$ of the universal $S$-Witt vectors group $W$, by the group of units ${\mathcal{O}}_S[[T]{]}^{\ast }$. We associate an $S$-functor ${\Im }_{\text{o}mb}$ to $\Im$, and we define an Abel–Jacobi morphism $f:\mathfrak{A}=\mathrm{Spec}A[[T]][T^{-1}],⟶{\Im }_{\text{o}mb}$ , 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 $\Im$ to its own Cartier dual group $\check{\Im }=\underline{\mathrm{Hom}}(\Im ;{\mathbb{G}}_m)$, and inducing the above isomorphism for $G={\mathbb{G}}_m$. It follows that $\Im$ may be interpreted as the relative Loop Group:

$S^{\prime }=\mathrm{Spec}{A}^{\prime }$ denotes a $S$-scheme, and we write ${\mathfrak{A}}_{\{S^{\prime }\}}=\mathrm{Spec}{A}^{\prime }[[T]][T^{-1}]$, and as the $A$-universal group of Witt-Bivectors.

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