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

Contou-Carrère, Carlos

C. R. Acad. Sci., Paris, Sér. I 318, No. 8, 743-746 (1994).

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