Cycle class map
In algebraic geometry, the Chow groups (named after W. L. Chow by Chevalley (1958)) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties (so-called algebraic cycles) in a similar way to how simplicial or cellular homology groups are formed out of subcomplexes. When the variety is smooth, the Chow groups can be interpreted as cohomology groups (compare Poincare duality) and have a multiplication called the intersection product. The Chow groups carry rich information about an algebraic variety, and they are correspondingly hard to compute in general.
For what follows, define a variety over a field k to be an integral scheme of finite type over k. For any scheme X of finite type over k, an algebraic cycle on X means a finite linear combination of subvarieties of X with integer coefficients. (Here and below, subvarieties are understood to be closed in X, unless stated otherwise.) For a natural number i, the group Zi(X) of i-dimensional cycles (or i-cycles, for short) on X is the free abelian group on the set of i-dimensional subvarieties of X.
For a variety W of dimension i + 1 and any rational function f on W which is not identically zero, the divisor of f is the i-cycle
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