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:
“Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
—Walter Pater (18391894)
“Mothers often are too easily intimidated by their childrens negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)
“Interpreting the dance: young women in white dancing in a ring can only be virgins; old women in black dancing in a ring can only be witches; but middle-aged women in colors, square dancing...?”
—Mason Cooley (b. 1927)