Let $X$ and $Y$ be varieties over $\mathbb{C}$, assumed projective and smooth. Suppose that $f : X \to Y$ is flat morphism. For any subscheme $W$ denote its fundamental class by $[W]$.
Then for any subvariety $Z \subset Y$, we can compute the scheme theoretic preimage $f^{-1}(Z)$. It can be non-reduced, non-irreducible in general, but it defines a homology class in $X$ by linearity (reduced varieties define fundamental classes by triangulation). Denote this
One can also find a cohomology class in $Y$ that is dual to the cohomology class of $Z$, $PD[Z]$ (poincare dual).
The following should be true, I hope:
$PD (f^* PD ([Z])) = [f^{-1}(Z)]$.
I am lost for how to prove this, even in the case that $Z$ is a point.
Flatness is necessary, I think, since for the map $\pi: Bl_p P^2 \to P^2$, $\pi^{-1}(p)$ and $\pi^{-1}(q)$ have different classes in homology. (For example because their self intersection numbers are different, and the intersection pairing is defined on homology.)