Definition of The Chow Ring
It is part of the definition of rational equivalence that it only holds between subvarieties of equal dimension. For the purposes of constructing the Chow ring, we are interested in the codimension of the subvariety (that is, the difference between its dimension and that of X) since it makes the product work properly, so we define the groups Ak(X), for integers k satisfying, to be the abelian group of formal sums of subvarieties of X of codimension k modulo rational equivalence. The Chow ring itself is the direct sum of these, namely,
The ring structure is given by intersection of varieties: that is, if we have two classes in Ak(X) and Al(X) respectively, we define their product to be
This definition has a number of technicalities that will be discussed below; here it suffices to say that in the best case, which can be shown always to hold up to rational equivalence, this intersection has codimension k + l, hence lies in Ak + l(X). This makes the Chow ring into a graded ring. As a matter of notation, an element of the Chow ring is often called a "cycle".
Read more about this topic: Chow Ring
Famous quotes containing the words definition of, definition and/or ring:
“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)
“... 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)
“He will not idly dance at his work who has wood to cut and cord before nightfall in the short days of winter; but every stroke will be husbanded, and ring soberly through the wood; and so will the strokes of that scholar’s pen, which at evening record the story of the day, ring soberly, yet cheerily, on the ear of the reader, long after the echoes of his axe have died away.”
—Henry David Thoreau (1817–1862)