Intersection multiplicity of Serre in the unramified case

We describe here some recent progress pertaining to the Serre Intersection Multiplicity Conjecture. In particular, we show that if A is unramified, then just as in the equicharacteristic case, the intersection multiplicity of two modules is bounded below by the product of their Hilbert–Samuel multiplicities. We also explain, in terms of the blowup of Spec A, the geometric significance of achieving this lower bound.