Lines crossing a tetrahedron and the Bloch group

  • Kevin Hutchinson

    University College Dublin, Ireland
  • Masha Vlasenko

    Trinity College, Dublin, Ireland
Lines crossing a tetrahedron and the Bloch group cover
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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 kk 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 CH2(Spec(k),3)\mathrm{CH}^2(\operatorname{Spec} (k),3) using only linear subvarieties in affine spaces and show that it maps surjectively to the Bloch group B(k)B(k) for any infinite field kk. We also describe the kernel of this map.