Jacobienne locale d'une courbe formelle relative

  • Carlos Contou-Carrère

    Université de Montpellier II, France


This article is devoted to the proof of a relative duality formula on a noetherian scheme , giving rise on the spectrum of a field to local symbols of class field theory. Relative local symbols are obtained in terms of the universal property of a couple , of a -group functor , associated to a -formal curve locally of the form (. is a -group extension of the completion of the universal -Witt vectors group , by the group of units . We associate an -functor to , and we define an Abel–Jacobi morphism , setting up a group isomorphism:

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

which is a local symbol (The Tame Symbol), identifying to its own Cartier dual group , and inducing the above isomorphism for . It follows that may be interpreted as the relative Loop Group:

denotes a -scheme, and we write , and as the -universal group of Witt-Bivectors.

The couple may be seen as the local analogue of the relative Rosenlicht Jacobian (Generalized Jacobian) defined by a -smooth curve .

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