The limit and boundary characteristic classes in Borel-Moore motivic homology

We show that the zero-dimensional part of the pro-Chern-Schwarz-MacPherson class defined by Aluffi can be lifted to the zeroth Suslin homology.

The proof uses the pro-characteristic class in the limit Borel-Moore motivic homology, which has a quadratic refinement in the limit Borel-Moore Milnor-Witt homology.

In characteristic zero, this construction factors through the group of constructible functions, in a way compatible with the covariant functoriality; in positive characteristic this property fails, and we show that the failure can be measured by the boundary characteristic class in the boundary Borel-Moore motivic homology.

We prove a push-forward formula for the boundary characteristic class, and conjecture it to agree with the Swan class defined by Kato-Saito.