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.



Cohomology of resolution of singularity
In general, the answer is no. It already fails for a nodal curve. In fancier terms, you can understand the obstruction as follows: if $X$ has a resolution $\tilde X$, and the cohomology of $X$ injects the cohomology of $\tilde X$, then $H^i(X)$ would be pure of weight $i$ as a Galois module/mixed Hodge structure (when over $\mathbb{C}$).


