Higher Intersection Theory on Algebraic Stacks I, by Roy Joshua

In this paper and the sequel we establish a theory of Chow groups and higher Chow groups on algebraic stacks of finite type over a field and establish their basic properties. This includes algebraic stacks in the sense of Deligne-Mumford as well as Artin. The main results of Part I are the following. The higher Chow groups are defined in general with respect to an atlas, but are shown to be independent of the choice of the atlas for smooth stacks if one uses finite coefficients with torsion prime to the characteristics or in general for Deligne-Mumford stacks. We also show there exist long exact localization sequences of the higher Chow groups modulo torsion for all Artin stacks and that these higher Chow groups modulo torsion are covariant for all representable proper maps that are locally projective while being contravariant for all representable flat maps. As an application of our theory we compute the higher Chow groups of Deligne-Mumford stacks and show that they are isomorphic modulo torsion to the higher Chow groups of their coarse moduli spaces. As a by-product of our theory we also produce localization sequences in (integral) higher Chow groups for all schemes of finite type over a field: these higher Chow groups are defined as the Zariski hypercohomology with respect to the cycle complex.


Roy Joshua <joshua@math.ohio-state.edu>