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.
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3$\begingroup$ A sufficient condition is that $\varphi$ is finite and surjective, for instance. But maybe you should be more specific on the reason why you want this kind of property. $\endgroup$– D.-C. CisinskiCommented Jan 20, 2012 at 21:34
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$\begingroup$ I need this property since I am considering morphisms between curves and the relationship between their cohomology groups. I computed some examples and found that this is injective. So I am wondering if this is true. Since in the case of curves, the morphisms are finite surjective, so you mentioned this is always true. This simplifies the computation. But do you know a reference of this, or the argument is easy? $\endgroup$– user565739Commented Jan 20, 2012 at 21:50
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3$\begingroup$ Since you're working in characteristic $0$, you might as well assume that you're over $\mathbb{C}$ and work with singular cohomology. In this case, if you stare at the Leray spectral sequence for $\varphi$ with constant coefficients, you find that the following two things would do the job for you: 1) The spectral sequence degenerates on the second page (this is the case, for example, if $f$ is finite). 2) The natural map $\mathbb{C}\to \varphi_*\mathbb{C}$ admits a section $\varphi_*\mathbb{C}\to\mathbb{C}$ (this is true, I believe, when $\varphi$ is generically etale). $\endgroup$– Keerthi MadapusiCommented Jan 20, 2012 at 23:26
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