If $X,Y$ are smooth projective schemes, then if we have a surjection $f:X\to Y$, we have an injective map on étale cohomology, or more generally on any Weil cohomology (see https://mathoverflow.net/q/172527). The proof of this statement uses Poincare duality on the target $Y$.

I would like to understand the cohomology of the resolution of singularity of a scheme $X$. If $\tilde{X}\to X$ is a resolution of singularites, then this is a surjective map (since it is a birational map between projective schemes). Can this be generalized to say that $$H^*(X)\to H^*(\tilde{X})$$ is injective? Since the ``usual" argument uses Poincare duality on the target $X$, and $X$ is non-smooth (otherwise, what's the point of the resolution of singularities), the usual argument fails.