I heard an algebraic geometry professor say that the Chow ring is usually isomorphic to the homology ring for cases we care about in application. However, I cannot find many results about this, beyond that it is isomorphic for $\mathbb{P}^n$. Is it isomorphic for things like products and blow-ups? What is the most general statement that I can use?
I am especially interested in blow-ups. I would like to understand the negative self-intersection of the exceptional divisor (i.e. in the blow up of $\mathbb{C}P^2$ at a point) but I want to use the "differential geometry" proof that goes by integrating the first Chern class, which should be on the topology side.