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.