Borel-moore Homology, Riemann-roch Transformations, and Local Terms

1.1. For a finite type separated Deligne-Mumford stack X/k and integer i, the i-th `-adic Borel-Moore homology group of X, denoted Hi(X), is defined to be H (X,ΩX), where ΩX ∈ D c(X,Q`) is the `-adic dualizing complex of X. These groups were considered already by Laumon in [17] (and, we have been informed, by Grothendieck in unpublished work), where he showed they enjoyed a number of good properties. In particular, there is a cycle class map