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:
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)
“... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
—Adrienne Rich (b. 1929)
“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)