2020/02/14 by Yuri Shimizu, Shimizu, Yuri
Mathematics · #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology #math.AG #math.KT
paper · pdf · doi:10.48550/arxiv.2002.05918
arxiv created 2020/05/25 · arxiv updated 2020/05/26
Let k be a field admitting a resolution of singularities. In this paper, we prove that the functor of zeroth \mathbbA1-homology H^\mathbbA10 is universal as a functorial birational invariant of smooth proper k-varieties taking values in a category enriched by abelian groups. For a smooth proper k-variety X, we also prove that the dimension of H^\mathbbA10(X;ℚ)(Spec k) coincides with the number of R-equivalence classes of X(k). We deduce these results as consequences of the structure theorem that for a smooth proper k-variety X, the sheaf H^\mathbbA10(X) is the free abelian presheaf generated by the birational \mathbbA1-connected components π0^b\mathbbA1(X) of Asok-Morel.