  We consider the operation of intersecting with a locally principal Cartier
divisor (i.e., a Cartier divisor which is principal on some neighborhood of its
support). We describe this operation explicitly on the level of cycles and
rational equivalences and as a corollary obtain a formula for rational
equivalence between intersections of two locally principal Cartier divisors.
Such canonical rational equivalence applies quite naturally to the setting of
algebraic stacks. We present two applications: (i) a simplification of the
development of Fulton-MacPherson-style intersection theory on Deligne-Mumford
stacks, and (ii) invariance of a key rational equivalence under a certain group
action (which is used in developing the theory of virtual fundamental classes
via intrinsic normal cones).
