Lines crossing a tetrahedron and the Bloch group

Abstract

According to B. Totaro (Milnor K-theory is the simplest part of algebraic K-theory, K-theory 6 (1992), 177–189), there is a hope that the Chow groups of a field $k$ can be computed using a very small class of affine algebraic varieties (linear spaces in the right coordinates), whereas the current definition uses essentially all algebraic cycles in affine space. In this note we consider a simple modification of ${\mathrm{CH}}^2(\mathrm{Spec}(k),3)$ using only linear subvarieties in affine spaces and show that it maps surjectively to the Bloch group $B(k)$ for any infinite field $k$. We also describe the kernel of this map.