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:
“The audience is the most revered member of the theater. Without an audience there is no theater. Every technique learned by the actor, every curtain, every flat on the stage, every careful analysis by the director, every coordinated scene, is for the enjoyment of the audience. They are our guests, our evaluators, and the last spoke in the wheel which can then begin to roll. They make the performance meaningful.”
—Viola Spolin (b. 1911)
“Our deeds disguise us. People need endless time to try on their deeds, until each knows the proper deeds for him to do. But every day, every hour, rushes by. There is no time.”
—Haniel Long (1888–1956)