Gillet in K-theory handbook (page 259) mentions that Roberts (The vanishing of intersection multiplicities on perfect complexes) uses Fulton’s operational Chow groups, which give an alternative method of constructing the product on rational Chow groups, for a general regular Noetherian scheme.
Gillet: K-theory and Intersection theory, Section 6.
These are a suggestion for a so called Chow cohomology theory. However, they are not homotopy invariant. See Gillet for more details.
arXiv: Experimental full text search
AG (Algebraic geometry)
Mixed
nLab page on Operational Chow groups