A theorem on smoothness - Bass-Quillen, Chow groups and intersection multiplicity of Serre

We describe here an inherent connection of smoothness among the Bass-Quillen conjecture, the Chow-group problem and Serre's Theorem on Intersection Multiplicity. Extension of a theorem of Lindel on smoothness plays a key role in our proof of the Serre-multiplicity theorem in the geometric (resp. unramified) case. We reduce the complete case of the theorem to the above case by using Artin's Approximation. We do not need the concept of "complete Tor". Similar proofs are sketched for Quillen's theorem on Chow groups and its extension due to Gillet and Levine.