Algebraic Cycle - Flat Pullback and Proper Pushforward

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.