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We describe here some recent progress pertaining to the Serre IntersectionMultiplicity Conjecture. In particular, we show that if A is an unramifiedregular local ring, then just as in the equicharacteristic case, theintersection multiplicity of two complimentary-dimensional modules is boundedbelow by the product of their Hilbert-Samuel multiplicities. We also explain,in terms of the blowup of Spec(A) at the closed point, the geometricsignificance of achieving this lower bound.

Intersection Multiplicity of Serre in the Unramified Case