  In the present paper, we define a notion of numerical equivalence on Chow
groups or Grothendieck groups of Noetherian local rings, which is an analogue
of that on smooth projective varieties. Under a mild condition, it is proved
that the Chow group modulo numerical equivalence is a finite dimensional ${\Bbb
Q}$-vector space, as in the case of smooth projective varieties. Numerical
equivalence on local rings is deeply related to that on smooth projective
varieties. For example, if Grothendieck's standard conjectures are true, then a
vanishing of Chow group (of local rings) modulo numerical equivalence can be
proven. Using the theory of numerical equivalence, the notion of numerically
Roberts rings is defined. It is proved that a Cohen-Macaulay local ring of
positive characteristic is a numerically Roberts ring if and only if the
Hilbert-Kunz multiplicity of a maximal primary ideal of finite projective
dimension is always equal to its colength. Numerically Roberts rings satisfy
the vanishing property of intersection multiplicities. We shall prove another
special case of the vanishing of intersection multiplicities using a vanishing
of localized Chern characters.
