Let $X/k$ and $Y/k$ be two smooth affine varieties over a field $k$ with $\mathrm{char}(k) = 0$ and $\varphi: X \rightarrow Y$ be a morphism. I would like to know under what conditions, the induced map $\varphi^{\ast}: H^i_{dR}(Y/k) \rightarrow H^i_{dR}(X/k) $ is injective. If $\varphi$ is dominant, then this is true for $i=0$. But for $i \geq 1$, I don't have idea.