Specialization of Zero Cycles

Let X be a proper scheme over a field K. There are two ways of organizing the points of X into equivalence classes using rational curves. One is the notion of R-equivalence introduced by [Manin72]. Two points x1, x2 ∈ X(K) are called directly R-equivalent if there is a morphism p : P → X with p(0:1) = x1 and p(1:0) = x2. This generates an equivalence relation called R-equivalence. The set of R-equivalence classes X(K)/R forms a set. Closely related to it is rational equivalence. Here we allow pairs of morphisms h : C → P and p : C → X and declare p∗(h(0:1)) and p∗(h (1:0)) to be rationally equivalent. Rational equivalence is sometimes hard to see geometrically. The set of rational equivalence classes forms a group CH0(X). Let S be the spectrum of a local Dedekind ring with residue field k and quotient field K. Let XS → SpecS be a proper morphism, xK a closed point of XK and xS its closure in XS. Then xS ∩ Xk is a zero cycle on Xk. This defines a specialization map on zero cycles. It is easy to prove (see, for instance, [Fulton84, 2.3]), that this descends to specialization maps