 Closed integral subschemes of "X" are in one-to-one correspondence with the scheme-theoretic points of "X" under the map that, in one direction, takes each subscheme to its generic point, and in the other direction, takes each point to the unique reduced subscheme supported on the closure of the point.Consequently formula_8 can also be described as the free abelian group on the points of "X".A cycle formula_9 is rationally equivalent to zero, written formula_10, if there are a finite number of formula_11-dimensional subvarieties formula_12 of formula_13 and non-zero rational functions formula_14 such that formula_15, where formula_16 denotes the divisor of a rational function on "W""i".