The Bloch-Quillen formula, especially a version with Milnor K-coefficients, makes it possible to express the product of the Chow ring through an ordinary cup product in sheaf cohomology and the concatenation product of symbols. No special care is needed if cycles do not intersect properly, no moving lemma nor deformation to the normal cone. We give an explicit formula for the intersection form along this line, different from the Serre Tor-formula.

This work has been supported by the DFG SFB/TR45. Periods, moduli spaces, and arithmetic of algebraic varieties.