Higher Intersection Theory on Algebraic Stacks: I

In this paper and the sequel we establish a theory of Chow groups and higher Chow groups on algebraic stacks locally 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. An intrinsic difference between our approach and earlier approaches is that the higher Chow groups of Bloch enter into our theory early on and depends heavily on his fundamental work. Our theory may be more appropriately called the (Lichtenbaum) motivic homology and cohomology of algebraic stacks. One of the main themes of these papers is that such a motivic homology does provide a reasonable intersection theory for algebraic stacks (of finite type over a field), with several key properties holding integrally and extending to stacks locally of finite type. While several important properties of our higher Chow groups, like covariance for projective representable maps (that factor as the composition of a closed immersion into the projective space associated to a locally free coherent sheaf and the obvious projection), an intersection pairing and contravariant functoriality for all smooth algebraic stacks, are shown to hold integrally, our theory works best with rational coefficients.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. (Using some results on motivic cohomology, we extend this integrally to all smooth algebraic stacks in Part II.) Using cohomological descent, we extend Bloch's fundamental localization sequence for quasi-projective schemes to long exact localization sequences of the higher Chow groups modulo torsion for all Artin stacks: this is one of the main results of the paper. We show that these higher Chow groups modulo torsion are covariant for all proper representable maps between stacks of finite type while being contravariant for all representable flat maps and, in Part II, that they are independent of the choice of an atlas for all stacks of finite type over the given field k. The comparison with motivic cohomology, as is worked out in Part II, enables us to provide an explicit comparison of our theory for quotient stacks associated to actions of linear algebraic groups on quasi-projective schemes with the corresponding Totaro–Edidin–Graham equivariant intersection theory. 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 locally of finite type over a field: these higher Chow groups are defined as the Zariski hypercohomology with respect to the cycle complex.