Let $k$ be a field, let $f \colon X \to Y$ be a morphism of $k$-varieties, and assume $X$ and $Y$ are smooth and projective. Let $H(\_)$ be a classical Weil cohomology theory (i.e. one of $\ell$-adic étale, Betti, algebraic de Rham, crystalline). Of course it depends on $k$ which of the theories are applicable, but I do not want to spell all the cases out here.
What conditions can one impose on $f$ to assert that $f^{*} \colon H^{i}(Y) \to H^{i}(X)$ is injective/surjective?
Currently I can only think of the rather trivial: if $f$ admits a section (resp. retraction), then $f^{*}$ is injective (resp. surjective).
[Edit] For example, is it true that $f^{*}$ is injective if $f$ is dominant? [/Edit]
Best Answer
Kleiman, Algebraic Cycles and the Weil Conjectures, Proposition 1.2.4: Let $f: X \to Y$ be surjective. Then $f^*: H^*(Y) \to H^*(X)$ is injective.
Let me recall the proof.
Let $x$ be a closed point of the generic fibre of $f$ and set $Z := \overline{\{z\}} \subseteq X$ and let $z = cl(Z)$ be the cycle class of $Z$. Since the cycle class map commutes with $f_*$, one has $f_*(z) \neq 0$.
Now assume $a \in H^*(Y)$ is in the kernel of $f^*$. Then one has $f^*(ab)z = f^*(a)f^*(b)z = 0$ for every $b \in H^*(Y)$, so by the projection formula $0 = f_*(f^*(ab)z) = abf_*(z)$, so by Poincaré duality $a = 0$.