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
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And the flat Earth becomes a Ball.”
—William Blake (1757–1827)
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—Thomas Hobbes (1579–1688)