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
Famous quotes containing the word definition:
“It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
—François, Duc De La Rochefoucauld (1613–1680)
“It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
—Jane Adams (20th century)
“The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (1772–1834)