Let $ X $ be a scheme of finite type over $k$.
Let $ \mathrm{CH}^i(X)$ be the Chow group of $ \ i \ $ cocycles on $X$.
- Is $ X \to \mathrm{CH}^i (X) $ a covariant or a contravariant functor ? Why ?
Let $ \mathrm{CH}_i(X)$ be the Chow group of $ \ i \ $ cycles on $X$.
- Is $ X \to \mathrm{CH}_i (X) $ a covariant or a contravariant functor ? Why ?
Thanks in advance for your help.
The pullback of a divisor is not always well-defined. For example, if $f : X \to \Bbb P^1$ is a constant map, then the pullback of a point could be $[X]$ or $0$. But if $f : X \to Y$ is a flat morphism then there is a pullback map $f^*$ which is contravariant.
On the other hand the pushforward is better behaved (we still ask $f$ to be proper). The definition is a bit tricky so I write it here for convenience : if $f : X \to Y$ and $Z \subset X$ is a closed subvariety then we define $f_*([Z]) := 0$ if $\dim f(Z) < \dim Z$, $n [f(Z)]$ if $\dim f(Z) = \dim Z$ and the restriction $f_{|Z} : Z \to f(Z)$ has degree $n$, and extend by linearity. This time this is covariant.
See 1.3.6, chapter on the Chow ring, section "Functoriality" in the book by Eisenbud and Harris, "3264 and all that". If you are looking for proof I think you need to look at the book by Fulton on intersection theory.