K-theory and intersection theory revisited (Q1106277)

If X is a nonsingular variety over k the Bloch-Quillen isomorphism \(\eta: CH\;p(X)\simeq H\;p(X,{\mathcal K}_ p({\mathcal O}_ X))\) describes the Chow group of codimension p cycles modulo rational equivalence in terms of sheaf cohomology. The following theorem is proved: ``Let X be a smooth variety over k, Y, Z two integral subschemes of codimension p and q, respectively, which intersect properly on X. If \(Y\cdot Z\) is the intersection cycle, then \(\eta (Y\cdot Z)=(-1)^{pq}\eta (Y)\eta (Z)\in H^{p+q}(X,{\mathcal K}_{p+q}({\mathcal O}_ X))9.\) Unlike the former proofs of Grayson and of the author, this proof is essentially a formal argument using natural properties of Quillen's spectral sequence, the K-theory product, cycle classes and the classical intersection product. It is obtained as a corollary that the Bloch-Quillen isomorphism is compatible with inverse images.