### 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>