## Local jacobian, universal Witt bivector group and the tame symbol. (Jacobienne locale, groupe de bivecteurs de Witt universel, et symbole modéré.)(French)Zbl 0840.14031

Summary: Let $$S= \text{Spec} (A)$$ be a noetherian affine scheme, $${\mathcal X}\to S$$ and $$S$$-formal curve isomorphic to $$\text{Spf} ({\mathcal O}_S [[T]])$$, and $${\mathcal U}= \text{Spec} (A[[T]][T^{-1}])$$. Let $$G$$ be any commutative, smooth and separated $$S$$-group scheme. We construct an $$S$$-group extension $${\mathcal F}$$ of the completion $$\check W$$, of the universal $$S$$-Witt vector group $$W$$, by the group of units $${\mathcal O}_S [[T]]^*$$, we associate an $$S$$-functor $${\mathcal F}_{\text{omb}}$$ to $${\mathcal F}$$, we define an Abel-Jacobi morphism $$f: {\mathcal U}\to {\mathcal F}_{\text{omb}}$$, which sets up an isomorphism $\operatorname{Hom}_{S\text{-gr}} ({\mathcal F},G) \overset\sim \rightarrow G({\mathcal U}). \tag $$*$$$ We define an $$S$$-bihomomorphism $${\mathcal F}\times {\mathcal F}\to \underline {\mathbb{G}}_{mS}$$, identifying $${\mathcal F}$$ to its own dual group, and inducing the isomorphism $$(*)$$ if $$G= \underline {\mathbb{G}}_{mS}$$.

### MSC:

 14L05 Formal groups, $$p$$-divisible groups 14L15 Group schemes 14G20 Local ground fields in algebraic geometry 13K05 Witt vectors and related rings (MSC2000)

### Keywords:

formal curve; Witt vector group; group scheme