A direct proof of the Steinberg property of the Contou-Carrère symbol

The aim of this work is to show that, when A is an artinian local ring, the Contou-Carrère symbol satisfies the property of Steinberg symbols: 〈f,1−f〉A(×(t))=1 for all elements f,1−f∈A(×(t)). Moreover, we offer a cohomological characterization of the Contou-Carrère symbol from the commutator of a central extension of groups.