Flat Pullback and Proper Pushforward
There is a covariant and a contravariant functoriality of the group of algebraic cycles. Let f : X → X' be a map of varieties.
If f is flat of some constant relative dimension (i.e. all fibers have the same dimension), we can define for any subvariety Y' ⊂ X':
which by assumption has the same codimension as Y′.
Conversely, if f is proper, for Y a subvariety of X the pushforward is defined to be
where n is the degree of the extension of function fields if the restriction of f to Y is finite and 0 otherwise.
By linearity, these definitions extend to homomorphisms of abelian groups
(the latter by virtue of the convention) are homomorphisms of abelian groups. See Chow ring for a discussion of the functoriality related to the ring structure.
Read more about this topic: Algebraic Cycle
Famous quotes containing the words flat and/or proper:
“Maybe Rose, there is always another story,
better unsaid, grim or flat or predatory.”
—Anne Sexton (1928–1974)
“The Creator has not thought proper to mark those in the forehead who are of stuff to make good generals. We are first, therefore, to seek them blindfold, and then let them learn the trade at the expense of great losses.”
—Thomas Jefferson (1743–1826)