Quillen’s K-theory and algebraic cycles on almost nonsingular varieties.Illinois J. Math. 25 (1981), no. 4, 654–666.

The author calls an irreducible algebraic variety X with just one singular point x0 “almost nonsingular”. The graded ring B(X) = Hi(X,Ki) (where Ki = KiOX is the sheaf of K-groups on X) is isomorphic to the Chow ring of algebraic cycles modulo linear equivalence when X is nonsingular. One interesting problem is to find a geometric or cycle-theoretic interpretation of B(X) when X is singular. Very little is known in general, except that B0(X) = Z and B1(X) = PicX. The author finds a simple description for B(X) when X is almost nonsingular, namely that Bi(X) is the group of cycles of codimension i missing the singular locus, modulo the subgroup generated by those cycles appearing as the divisor of some rational function on some subvariety of codimension i−1 missing the singular locus. He calls the latter group CHi(X, x0). This description is simple and surprising, and the author’s proof does not seem to generalize.