Intersection theory on weighted blow-ups

I'll start with a short overview of some classical results for blow-ups. When looking at the case of weighted blow-ups, some caution is needed as we don't have access to all of the tools used in the classical case. One needs, for example, to take into account that the weights make for some non reducedness or some complications with torsion in the Chow rings. I will give a general presentation for the Chow ring of the exceptional divisor of a weighted blow-up and for the Gysin homomorphism induced by it.