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Concept
Chow group
Summary
In algebraic geometry, the Chow groups (named after Wei-Liang Chow by ) 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 Poincaré 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 to be an integral scheme of finite type over . For any scheme of finite type over , an algebraic cycle on means a finite linear combination of subvarieties of with integer coefficients. (Here and below, subvarieties are understood to be closed in , unless stated otherwise.) For a natural number , the group of -dimensional cycles (or -cycles, for short) on is the free abelian group on the set of -dimensional subvarieties of .
For a variety of dimension and any rational function on which is not identically zero, the divisor of is the -cycle
where the sum runs over all -dimensional subvarieties of and the integer denotes the order of vanishing of along . (Thus is negative if has a pole along .) The definition of the order of vanishing requires some care for singular.
For a scheme of finite type over , the group of -cycles rationally equivalent to zero is the subgroup of generated by the cycles for all -dimensional subvarieties of and all nonzero rational functions on . The Chow group of -dimensional cycles on is the quotient group of by the subgroup of cycles rationally equivalent to zero. Sometimes one writes for the class of a subvariety in the Chow group, and if two subvarieties and have , then and are said to be rationally equivalent.
For example, when is a variety of dimension , the Chow group is the divisor class group of .
Official source
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