Bloch group

Definition

Let $F$ be a field. Let $\mathbb{Z}[F]$ be the free abelian group on symbols $[x]$ for $x \in F \setminus \{0, 1\}$. The pre-Bloch group $\mathcal{P}(F)$ is the quotient of $\mathbb{Z}[F]$ by the subgroup generated by the five-term elements $$[x] - [y] + \Big[\frac{y}{x}\Big] - \Big[\frac{1 - x^{-1}}{1 - y^{-1}}\Big] + \Big[\frac{1 - x}{1 - y}\Big]$$ for all $x, y \in F \setminus \{0, 1\}$ with $x \ne y$. Let $\Lambda^2_\Z F^\times$ be the second exterior power over $\Z$ of the unit group $F^\times$, written additively. The map $\delta \colon \mathcal{P}(F) \to \Lambda^2_\Z F^\times$, $[x] \mapsto x \wedge (1 - x)$, is a well-defined homomorphism. The Bloch group of $F$ is its kernel: $$\mathcal{B}(F) = \ker\big(\delta \colon \mathcal{P}(F) \to \Lambda^2_\Z F^\times\big).$$