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:
“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)
“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)