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.*



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> What conditions can one impose on $f$ to assert that $f^{\*} \colon H^{i}(Y) \to H^{i}(X)$ is injective/surjective?
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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]**



What morphisms induce injective/surjective maps on (Weil) cohomology?
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 $a = 0$.


