Definition
An algebraic cycle of an algebraic variety or scheme X is a formal linear combination V = ∑ ni·Vi of irreducible reduced closed subschemes. The coefficient ni is the multiplicity of Vi in V. Initially the coefficients are taken to be integers, but rational coefficients are also widely used.
Under the correspondence
- {irreducible reduced closed subschemes V ⊂ X} ↭ {points of X}
(V maps to its generic point (with respect to the Zariski topology), conversely a point maps to its closure (with the reduced subscheme structure)) an algebraic cycle is thus just a formal linear combination of points of X.
The group of cycles naturally forms a group Z*(X) graded by the dimension of the cycles. The grading by codimension is also useful, then the group is usually written Z*(X).
Read more about this topic: Algebraic Cycle
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